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Linear_Algebra_In_Python

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1478 
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  7.   <title>Vectors And Spaces</title>
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    80. 

  80. <h1 id="vectors-and-spaces">Vectors And Spaces</h1>
    81. 

  81. <blockquote>
    82. 

  82. <p>by Thom Ives, Ph.D. using Python applied to lectures of Sal Khan from Khan Academy</p>
    83. 

  83. </blockquote>
    84. 

  84. <figure>
    85. 

  85. <img src="Khan_LA_Plus_Python_MT.png" alt="Khan and Python are MORE TOGETHER" /><figcaption>Khan and Python are MORE TOGETHER</figcaption>
    86. 

  86. </figure>
    87. 

  87. <p>In case you’ve never heard me say it, “I love Sal Khan’s teaching!” He is a gift to America and the world! When I need to review something in Linear Algebra, I usually turn to his videos and Khan Academy materials first. Well, a while back, a group of people wanted to learn Linear Algebra from me. I said,</p>
    88. 

  88. <blockquote>
    89. 

  89. <p>“Learn it from the best! And then I will help you learn how to use Python for linear algebraic operations and linear algebraic visualizations to help you learn it AND Python better!”</p>
    90. 

  90. </blockquote>
    91. 

  91. <p>Well, now I want to create a repo for this and share it with you as I grow it. I am eager to hear from you where I can make things more clear and helpful. This is a work in progress.</p>
    92. 

  92. <p>Here are the links we will cover below. Khan Academy Section www.khanacademy.org/math/linear-algebra/vectors-and-spaces 1. Vector Intro For Linear Algebra www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/vector-introduction-linear-algebra?modal=1 1. Real Coordinate Spaces www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/real-coordinate-spaces?modal=1 1. Adding Vectors Algebraically &amp; Graphically www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/adding-vectors 1. Multiplying A Vector By A Scalar www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/multiplying-vector-by-scalar 1. Vector Examples www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/linear-algebra-vector-examples?modal=1 1. Unit Vectors Introduction www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/intro-unit-vector-notation?modal=1 1. Parametric Representations Of Lines www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/linear-algebra-parametric-representations-of-lines?modal=1</p>
    93. 

  93. <p>Vectors have direction and magnitude.</p>
    94. 

  94. <p>It does NOT matter where they start from.</p>
    95. 

  95. <p>What are some vectors that you know about and what is their direction?</p>
    96. 

  96. <p>$  = [5 ] $ where <span class="math inline"><em>î</em></span> is a unit vector in the x direction.</p>
    97. 

  97. <div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb1-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    98. 

  98. <a class="sourceLine" id="cb1-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    99. 

  99. <a class="sourceLine" id="cb1-3" title="3"></a>
    100. 

  100. <a class="sourceLine" id="cb1-4" title="4">plt.rc_context({</a>
    101. 

  101. <a class="sourceLine" id="cb1-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    102. 

  102. <a class="sourceLine" id="cb1-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    103. 

  103. <a class="sourceLine" id="cb1-7" title="7"></a>
    104. 

  104. <a class="sourceLine" id="cb1-8" title="8">plt.plot([<span class="dv">2</span>, <span class="fl">2.001</span>],[<span class="dv">1</span>, <span class="fl">1.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    105. 

  105. <a class="sourceLine" id="cb1-9" title="9">plt.plot([<span class="fl">8.999</span>, <span class="dv">9</span>],[<span class="fl">3.999</span>, <span class="dv">4</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    106. 

  106. <a class="sourceLine" id="cb1-10" title="10"></a>
    107. 

  107. <a class="sourceLine" id="cb1-11" title="11">vi <span class="op">=</span> <span class="dv">5</span></a>
    108. 

  108. <a class="sourceLine" id="cb1-12" title="12">vj <span class="op">=</span> <span class="dv">0</span></a>
    109. 

  109. <a class="sourceLine" id="cb1-13" title="13">plt.arrow(<span class="dv">3</span>, <span class="dv">2</span>, vi, vj, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    110. 

  110. <a class="sourceLine" id="cb1-14" title="14">plt.show()<span class="op">;</span></a></code></pre></div>
    111. 

  111. <figure>
    112. 

  112. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_6_0.png" alt="png" /><figcaption>png</figcaption>
    113. 

  113. </figure>
    114. 

  114. <p>$  = [3 ;; 4] $ where <span class="math inline"><em>ĵ</em></span> is a unit vector in the y direction.</p>
    115. 

  115. <p>What is the magnitude of $  $?</p>
    116. 

  116. <p>It is $  =  =  =  = 5$</p>
    117. 

  117. <div class="sourceCode" id="cb2"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb2-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    118. 

  118. <a class="sourceLine" id="cb2-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    119. 

  119. <a class="sourceLine" id="cb2-3" title="3"></a>
    120. 

  120. <a class="sourceLine" id="cb2-4" title="4">plt.rc_context({</a>
    121. 

  121. <a class="sourceLine" id="cb2-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    122. 

  122. <a class="sourceLine" id="cb2-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    123. 

  123. <a class="sourceLine" id="cb2-7" title="7"></a>
    124. 

  124. <a class="sourceLine" id="cb2-8" title="8">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    125. 

  125. <a class="sourceLine" id="cb2-9" title="9">plt.plot([<span class="fl">7.999</span>, <span class="dv">8</span>],[<span class="fl">7.999</span>, <span class="dv">8</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    126. 

  126. <a class="sourceLine" id="cb2-10" title="10"></a>
    127. 

  127. <a class="sourceLine" id="cb2-11" title="11">vi <span class="op">=</span> <span class="dv">3</span></a>
    128. 

  128. <a class="sourceLine" id="cb2-12" title="12">vj <span class="op">=</span> <span class="dv">4</span></a>
    129. 

  129. <a class="sourceLine" id="cb2-13" title="13">plt.arrow(<span class="dv">1</span>, <span class="dv">2</span>, vi, vj, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    130. 

  130. <a class="sourceLine" id="cb2-14" title="14">plt.show()<span class="op">;</span></a></code></pre></div>
    131. 

  131. <figure>
    132. 

  132. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_8_0.png" alt="png" /><figcaption>png</figcaption>
    133. 

  133. </figure>
    134. 

  134. <h2 id="adding-two-vectors-of-like-kind">Adding Two Vectors Of Like Kind</h2>
    135. 

  135. <p>$  = [1 ;; 2], ;;;  = [2 ;; 2] $</p>
    136. 

  136. <p>It is $  =  +  = [(1+2) + (2+2)] = [3 + 4] $</p>
    137. 

  137. <p>What is the magnitude of $  $?</p>
    138. 

  138. <p>It is $  =  +  =  =  =  = 5$</p>
    139. 

  139. <div class="sourceCode" id="cb3"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb3-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    140. 

  140. <a class="sourceLine" id="cb3-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    141. 

  141. <a class="sourceLine" id="cb3-3" title="3"></a>
    142. 

  142. <a class="sourceLine" id="cb3-4" title="4">plt.rc_context({  <span class="co"># Only needed for some dark modes</span></a>
    143. 

  143. <a class="sourceLine" id="cb3-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    144. 

  144. <a class="sourceLine" id="cb3-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    145. 

  145. <a class="sourceLine" id="cb3-7" title="7"></a>
    146. 

  146. <a class="sourceLine" id="cb3-8" title="8"><span class="co"># Create a pallet plot for vectors</span></a>
    147. 

  147. <a class="sourceLine" id="cb3-9" title="9">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    148. 

  148. <a class="sourceLine" id="cb3-10" title="10">plt.xlim([<span class="dv">0</span>, <span class="dv">5</span>])<span class="op">;</span> plt.ylim([<span class="dv">0</span>, <span class="dv">7</span>])</a>
    149. 

  149. <a class="sourceLine" id="cb3-11" title="11"></a>
    150. 

  150. <a class="sourceLine" id="cb3-12" title="12">v1i <span class="op">=</span> <span class="dv">1</span><span class="op">;</span> v1j <span class="op">=</span> <span class="dv">2</span><span class="op">;</span> v2i <span class="op">=</span> <span class="dv">2</span><span class="op">;</span> v2j <span class="op">=</span> <span class="dv">2</span></a>
    151. 

  151. <a class="sourceLine" id="cb3-13" title="13"></a>
    152. 

  152. <a class="sourceLine" id="cb3-14" title="14">plt.arrow(<span class="dv">1</span>, <span class="dv">2</span>, v1i, v1j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    153. 

  153. <a class="sourceLine" id="cb3-15" title="15">          ec <span class="op">=</span><span class="st">&#39;green&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    154. 

  154. <a class="sourceLine" id="cb3-16" title="16">plt.arrow(<span class="dv">2</span>, <span class="dv">4</span>, v2i, v2j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    155. 

  155. <a class="sourceLine" id="cb3-17" title="17">          ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    156. 

  156. <a class="sourceLine" id="cb3-18" title="18"></a>
    157. 

  157. <a class="sourceLine" id="cb3-19" title="19">v3i <span class="op">=</span> v1i <span class="op">+</span> v2i<span class="op">;</span> v3j <span class="op">=</span> v1j <span class="op">+</span> v2j</a>
    158. 

  158. <a class="sourceLine" id="cb3-20" title="20">v3_mag <span class="op">=</span> np.linalg.norm([<span class="dv">3</span>, <span class="dv">4</span>])</a>
    159. 

  159. <a class="sourceLine" id="cb3-21" title="21"></a>
    160. 

  160. <a class="sourceLine" id="cb3-22" title="22">stuff <span class="op">=</span> <span class="vs">r&#39;The magnitude of $\vec</span><span class="sc">{v_3}</span><span class="vs">$&#39;</span> <span class="op">+</span> <span class="ss">f&#39; is </span><span class="sc">{</span>v3_mag<span class="sc">}</span><span class="ss">&#39;</span></a>
    161. 

  161. <a class="sourceLine" id="cb3-23" title="23">plt.text(<span class="dv">1</span>, <span class="dv">1</span>, stuff, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    162. 

  162. <a class="sourceLine" id="cb3-24" title="24">plt.arrow(<span class="dv">1</span>, <span class="dv">2</span>, v3i, v3j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    163. 

  163. <a class="sourceLine" id="cb3-25" title="25">          ec <span class="op">=</span><span class="st">&#39;blue&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    164. 

  164. <a class="sourceLine" id="cb3-26" title="26">plt.grid()</a>
    165. 

  165. <a class="sourceLine" id="cb3-27" title="27">plt.show()<span class="op">;</span></a></code></pre></div>
    166. 

  166. <figure>
    167. 

  167. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_10_0.png" alt="png" /><figcaption>png</figcaption>
    168. 

  168. </figure>
    169. 

  169. <p>How do you find the magnitude of a vector in NumPy?</p>
    170. 

  170. <p>You use norm - short for Euclidean Norm.</p>
    171. 

  171. <div class="sourceCode" id="cb4"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb4-1" title="1">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])</a>
    172. 

  172. <a class="sourceLine" id="cb4-2" title="2">v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">2</span>])</a>
    173. 

  173. <a class="sourceLine" id="cb4-3" title="3">v3 <span class="op">=</span> v1 <span class="op">+</span> v2</a>
    174. 

  174. <a class="sourceLine" id="cb4-4" title="4"><span class="bu">print</span>(v3)</a>
    175. 

  175. <a class="sourceLine" id="cb4-5" title="5">v3_mag <span class="op">=</span> np.linalg.norm(v3)</a>
    176. 

  176. <a class="sourceLine" id="cb4-6" title="6"><span class="bu">print</span>(v3_mag)</a></code></pre></div>
    177. 

  177. <pre><code>[3 4]
    178. 

  178. 5.0</code></pre>
    179. 

  179. <p>How do we scale vectors in NumPy as Sal Khan showed us?</p>
    180. 

  180. <div class="sourceCode" id="cb6"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb6-1" title="1">v4 <span class="op">=</span> <span class="dv">4</span><span class="op">*</span>v1</a>
    181. 

  181. <a class="sourceLine" id="cb6-2" title="2">v5 <span class="op">=</span> <span class="dv">4</span><span class="op">*</span>v2</a>
    182. 

  182. <a class="sourceLine" id="cb6-3" title="3"><span class="bu">print</span>(v4)</a>
    183. 

  183. <a class="sourceLine" id="cb6-4" title="4"><span class="bu">print</span>(v5)</a></code></pre></div>
    184. 

  184. <pre><code>[8 8]
    185. 

  185. [4 8]</code></pre>
    186. 

  186. <p>Now we can simply add our scaled vectors.</p>
    187. 

  187. <div class="sourceCode" id="cb8"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb8-1" title="1">v6 <span class="op">=</span> v4 <span class="op">+</span> v5</a>
    188. 

  188. <a class="sourceLine" id="cb8-2" title="2"><span class="bu">print</span>(v6)</a>
    189. 

  189. <a class="sourceLine" id="cb8-3" title="3">v6_mag <span class="op">=</span> np.linalg.norm(v6)</a>
    190. 

  190. <a class="sourceLine" id="cb8-4" title="4"><span class="bu">print</span>(v6_mag)</a></code></pre></div>
    191. 

  191. <pre><code>[12 16]
    192. 

  192. 20.0</code></pre>
    193. 

  193. <p>Can I find the unit vectors for all our vectors above? YES! Divide the vectors by their norms!</p>
    194. 

  194. <div class="sourceCode" id="cb10"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb10-1" title="1">v1_unit <span class="op">=</span> v1 <span class="op">/</span> np.linalg.norm(v1)</a>
    195. 

  195. <a class="sourceLine" id="cb10-2" title="2"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v1 is </span><span class="sc">{</span>v1_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    196. 

  196. <a class="sourceLine" id="cb10-3" title="3"></a>
    197. 

  197. <a class="sourceLine" id="cb10-4" title="4">v2_unit <span class="op">=</span> v2 <span class="op">/</span> np.linalg.norm(v2)</a>
    198. 

  198. <a class="sourceLine" id="cb10-5" title="5"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v2 is </span><span class="sc">{</span>v2_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    199. 

  199. <a class="sourceLine" id="cb10-6" title="6"></a>
    200. 

  200. <a class="sourceLine" id="cb10-7" title="7">v3_unit <span class="op">=</span> v3 <span class="op">/</span> v3_mag</a>
    201. 

  201. <a class="sourceLine" id="cb10-8" title="8"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v3 is </span><span class="sc">{</span>v3_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    202. 

  202. <a class="sourceLine" id="cb10-9" title="9"></a>
    203. 

  203. <a class="sourceLine" id="cb10-10" title="10">v4_unit <span class="op">=</span> v4 <span class="op">/</span> np.linalg.norm(v4)</a>
    204. 

  204. <a class="sourceLine" id="cb10-11" title="11"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v4 is </span><span class="sc">{</span>v4_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    205. 

  205. <a class="sourceLine" id="cb10-12" title="12"></a>
    206. 

  206. <a class="sourceLine" id="cb10-13" title="13">v5_unit <span class="op">=</span> v5 <span class="op">/</span> np.linalg.norm(v5)</a>
    207. 

  207. <a class="sourceLine" id="cb10-14" title="14"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v5 is </span><span class="sc">{</span>v5_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    208. 

  208. <a class="sourceLine" id="cb10-15" title="15"></a>
    209. 

  209. <a class="sourceLine" id="cb10-16" title="16">v6_unit <span class="op">=</span> v6 <span class="op">/</span> np.linalg.norm(v6)</a>
    210. 

  210. <a class="sourceLine" id="cb10-17" title="17"><span class="bu">print</span>(<span class="ss">f&#39;Unit vector of v6 is </span><span class="sc">{</span>v6_unit<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a></code></pre></div>
    211. 

  211. <pre><code>Unit vector of v1 is [0.70710678 0.70710678]
    212. 

  212. 

  213. Unit vector of v2 is [0.4472136  0.89442719]
    214. 

  214. 

  215. Unit vector of v3 is [0.6 0.8]
    216. 

  216. 

  217. Unit vector of v4 is [0.70710678 0.70710678]
    218. 

  218. 

  219. Unit vector of v5 is [0.4472136  0.89442719]
    220. 

  220. 

  221. Unit vector of v6 is [0.6 0.8]</code></pre>
    222. 

  222. <p>Why are they called unit vectors? If you find their norm, their norms will ALWAYS equal 1 IF they are truly unit vectors.</p>
    223. 

  223. <div class="sourceCode" id="cb12"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb12-1" title="1"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v1_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v1_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    224. 

  224. <a class="sourceLine" id="cb12-2" title="2"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v2_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v2_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    225. 

  225. <a class="sourceLine" id="cb12-3" title="3"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v3_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v3_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    226. 

  226. <a class="sourceLine" id="cb12-4" title="4"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v4_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v4_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    227. 

  227. <a class="sourceLine" id="cb12-5" title="5"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v5_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v5_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    228. 

  228. <a class="sourceLine" id="cb12-6" title="6"><span class="bu">print</span>(<span class="ss">f&#39;The magnitude of v6_unit is </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(v6_unit), <span class="dv">2</span>)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a></code></pre></div>
    229. 

  229. <pre><code>The magnitude of v1_unit is 1.0
    230. 

  230. 

  231. The magnitude of v2_unit is 1.0
    232. 

  232. 

  233. The magnitude of v3_unit is 1.0
    234. 

  234. 

  235. The magnitude of v4_unit is 1.0
    236. 

  236. 

  237. The magnitude of v5_unit is 1.0
    238. 

  238. 

  239. The magnitude of v6_unit is 1.0</code></pre>
    240. 

  240. <p>IF you multiply your unit vectors by a magnitude or scalar, you will get a vector in the same direction with a different magnitude.</p>
    241. 

  241. <h1 id="linear-combinations-and-span">Linear Combinations And Span</h1>
    242. 

  242. <p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-combinations/v/linear-combinations-and-span">Khan Academy Section</a></p>
    243. 

  243. <p>       <a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-combinations/v/linear-combinations-and-span?modal=1">Video for Linear Combinations And Span</a></p>
    244. 

  244. <p>$  = [1 ;; 2], ;;;  = [2 ;; 2] $ are independent.</p>
    245. 

  245. <p>We can say that some $  = c_1  + c_2  $</p>
    246. 

  246. <p>How much space could be spanned by using many different values for <span class="math inline"><em>c</em><sub>1</sub></span> and <span class="math inline"><em>c</em><sub>2</sub></span> ?</p>
    247. 

  247. <p>It would be all real values in 2 dimensional space.</p>
    248. 

  248. <p>In other words, $  ;; ;; ℝ^2 $.</p>
    249. 

  249. <p>The same can be true for independent vectors in $ ℝ^3 $.</p>
    250. 

  250. <p>$  = [1 ;; 2 ;; 3], ;;;  = [2 ;; 2 ;; 1] $ are independent.</p>
    251. 

  251. <p>We can say that some $  = c_1  + c_2  $</p>
    252. 

  252. <p>How much space could be spanned by using many different values for <span class="math inline"><em>c</em><sub>1</sub></span> and <span class="math inline"><em>c</em><sub>2</sub></span> ?</p>
    253. 

  253. <p>It would be all real values in 3 dimensional space.</p>
    254. 

  254. <p>In other words, $  ;; ;; ℝ^3 $.</p>
    255. 

  255. <p>Etcetera for more than 3 dimensions.</p>
    256. 

  256. <div class="sourceCode" id="cb14"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb14-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    257. 

  257. <a class="sourceLine" id="cb14-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    258. 

  258. <a class="sourceLine" id="cb14-3" title="3"></a>
    259. 

  259. <a class="sourceLine" id="cb14-4" title="4">c1 <span class="op">=</span> <span class="dv">3</span><span class="op">;</span> c2 <span class="op">=</span> <span class="dv">4</span></a>
    260. 

  260. <a class="sourceLine" id="cb14-5" title="5">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])<span class="op">;</span> v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">2</span>])</a>
    261. 

  261. <a class="sourceLine" id="cb14-6" title="6">v3 <span class="op">=</span> c1<span class="op">*</span>v1 <span class="op">+</span> c2<span class="op">*</span>v2</a>
    262. 

  262. <a class="sourceLine" id="cb14-7" title="7"><span class="bu">print</span>(v3)</a>
    263. 

  263. <a class="sourceLine" id="cb14-8" title="8">v3_mag <span class="op">=</span> <span class="bu">round</span>(np.linalg.norm(v3), <span class="dv">2</span>)</a>
    264. 

  264. <a class="sourceLine" id="cb14-9" title="9"></a>
    265. 

  265. <a class="sourceLine" id="cb14-10" title="10">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    266. 

  266. <a class="sourceLine" id="cb14-11" title="11">plt.xlim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])</a>
    267. 

  267. <a class="sourceLine" id="cb14-12" title="12">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    268. 

  268. <a class="sourceLine" id="cb14-13" title="13"></a>
    269. 

  269. <a class="sourceLine" id="cb14-14" title="14">stuff <span class="op">=</span> <span class="vs">r&#39;The magnitude of $\vec</span><span class="sc">{v_3}</span><span class="vs">$&#39;</span> <span class="op">+</span> <span class="ss">f&#39; is </span><span class="sc">{</span>v3_mag<span class="sc">}</span><span class="ss">&#39;</span></a>
    270. 

  270. <a class="sourceLine" id="cb14-15" title="15">plt.title(stuff, color<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    271. 

  271. <a class="sourceLine" id="cb14-16" title="16">plt.show()<span class="op">;</span></a></code></pre></div>
    272. 

  272. <pre><code>[10 14]</code></pre>
    273. 

  273. <figure>
    274. 

  274. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_23_1.png" alt="png" /><figcaption>png</figcaption>
    275. 

  275. </figure>
    276. 

  276. <h2 id="the-span-of-linearly-combined-independent-vectors">The Span of Linearly Combined Independent Vectors</h2>
    277. 

  277. <p>$ V = Span(, , , ) c_1 , c_2 , , c_3  ; ; c_i ℝ $</p>
    278. 

  278. <div class="sourceCode" id="cb16"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb16-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    279. 

  279. <a class="sourceLine" id="cb16-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    280. 

  280. <a class="sourceLine" id="cb16-3" title="3"></a>
    281. 

  281. <a class="sourceLine" id="cb16-4" title="4">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])<span class="op">;</span> v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">2</span>])</a>
    282. 

  282. <a class="sourceLine" id="cb16-5" title="5"></a>
    283. 

  283. <a class="sourceLine" id="cb16-6" title="6">finite_span <span class="op">=</span> <span class="dv">10</span></a>
    284. 

  284. <a class="sourceLine" id="cb16-7" title="7"><span class="cf">for</span> c1 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    285. 

  285. <a class="sourceLine" id="cb16-8" title="8">    <span class="cf">for</span> c2 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    286. 

  286. <a class="sourceLine" id="cb16-9" title="9">        v3 <span class="op">=</span> c1<span class="op">*</span>v1 <span class="op">+</span> c2<span class="op">*</span>v2</a>
    287. 

  287. <a class="sourceLine" id="cb16-10" title="10">        plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    288. 

  288. <a class="sourceLine" id="cb16-11" title="11">        plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    289. 

  289. <a class="sourceLine" id="cb16-12" title="12">                  width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    290. 

  290. <a class="sourceLine" id="cb16-13" title="13"></a>
    291. 

  291. <a class="sourceLine" id="cb16-14" title="14">plt.xlim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])</a>
    292. 

  292. <a class="sourceLine" id="cb16-15" title="15">plt.show()<span class="op">;</span></a></code></pre></div>
    293. 

  293. <figure>
    294. 

  294. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_25_0.png" alt="png" /><figcaption>png</figcaption>
    295. 

  295. </figure>
    296. 

  296. <h2 id="the-span-of-linearly-combined-dependent-vectors">The Span of Linearly Combined Dependent Vectors</h2>
    297. 

  297. <div class="sourceCode" id="cb17"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb17-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    298. 

  298. <a class="sourceLine" id="cb17-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    299. 

  299. <a class="sourceLine" id="cb17-3" title="3"></a>
    300. 

  300. <a class="sourceLine" id="cb17-4" title="4"></a>
    301. 

  301. <a class="sourceLine" id="cb17-5" title="5">finite_span <span class="op">=</span> <span class="dv">5</span></a>
    302. 

  302. <a class="sourceLine" id="cb17-6" title="6">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])<span class="op">;</span> v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">1</span>])</a>
    303. 

  303. <a class="sourceLine" id="cb17-7" title="7"></a>
    304. 

  304. <a class="sourceLine" id="cb17-8" title="8"><span class="cf">for</span> c1 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span <span class="op">+</span> <span class="dv">1</span>, <span class="dv">1</span>):</a>
    305. 

  305. <a class="sourceLine" id="cb17-9" title="9">    <span class="cf">for</span> c2 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span <span class="op">+</span> <span class="dv">1</span>, <span class="dv">1</span>):</a>
    306. 

  306. <a class="sourceLine" id="cb17-10" title="10">        v3 <span class="op">=</span> c1<span class="op">*</span>v1 <span class="op">+</span> c2<span class="op">*</span>v2</a>
    307. 

  307. <a class="sourceLine" id="cb17-11" title="11"></a>
    308. 

  308. <a class="sourceLine" id="cb17-12" title="12">        plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    309. 

  309. <a class="sourceLine" id="cb17-13" title="13">        plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    310. 

  310. <a class="sourceLine" id="cb17-14" title="14">                  width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    311. 

  311. <a class="sourceLine" id="cb17-15" title="15"></a>
    312. 

  312. <a class="sourceLine" id="cb17-16" title="16">plt.xlim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">20</span>, <span class="dv">20</span>])</a>
    313. 

  313. <a class="sourceLine" id="cb17-17" title="17">plt.show()<span class="op">;</span></a></code></pre></div>
    314. 

  314. <figure>
    315. 

  315. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_27_0.png" alt="png" /><figcaption>png</figcaption>
    316. 

  316. </figure>
    317. 

  317. <h1 id="linear-dependence-and-independence">Linear Dependence And Independence</h1>
    318. 

  318. <h3 id="linear-dependence-and-independence-1"><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence/v/linear-algebra-introduction-to-linear-independence">Linear Dependence And Independence</a></h3>
    319. 

  319. <ul>
    320. 

  320. <li><p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence/v/linear-algebra-introduction-to-linear-independence">Introduction To Linear Independence</a></p></li>
    321. 

  321. <li><p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence/v/more-on-linear-independence?modal=1">More On Linear Independence</a></p></li>
    322. 

  322. <li><p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence/v/span-and-linear-independence-example?modal=1">Span And Linear Independence Example</a></p></li>
    323. 

  323. </ul>
    324. 

  324. <h2 id="overview">Overview</h2>
    325. 

  325. <p>Linear Independence - No collinearity between two or more vectors (i.e. features) Linear Dependence - Collinearity is present between two or more vectors (i.e. features)</p>
    326. 

  326. <h2 id="linear-dependence">Linear Dependence</h2>
    327. 

  327. <div class="sourceCode" id="cb18"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb18-1" title="1">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])</a>
    328. 

  328. <a class="sourceLine" id="cb18-2" title="2">v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">1</span>])</a>
    329. 

  329. <a class="sourceLine" id="cb18-3" title="3"></a>
    330. 

  330. <a class="sourceLine" id="cb18-4" title="4"><span class="bu">print</span>(v1)</a>
    331. 

  331. <a class="sourceLine" id="cb18-5" title="5"><span class="bu">print</span>(v2)</a></code></pre></div>
    332. 

  332. <pre><code>[2 2]
    333. 

  333. [1 1]</code></pre>
    334. 

  334. <div class="sourceCode" id="cb20"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb20-1" title="1"><span class="bu">print</span>(v1)</a>
    335. 

  335. <a class="sourceLine" id="cb20-2" title="2"><span class="bu">print</span>(<span class="dv">2</span><span class="op">*</span>v2)</a></code></pre></div>
    336. 

  336. <pre><code>[2 2]
    337. 

  337. [2 2]</code></pre>
    338. 

  338. <p>This is easy to see by inspection, but what if we have many MANY dimensions? We need a mathematical way / a code way / a programmatic way to check for dependence or independence.</p>
    339. 

  339. <div class="sourceCode" id="cb22"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb22-1" title="1">u_v1 <span class="op">=</span> v1 <span class="op">/</span> np.linalg.norm(v1)</a>
    340. 

  340. <a class="sourceLine" id="cb22-2" title="2">u_v2 <span class="op">=</span> v2 <span class="op">/</span> np.linalg.norm(v2)</a>
    341. 

  341. <a class="sourceLine" id="cb22-3" title="3"></a>
    342. 

  342. <a class="sourceLine" id="cb22-4" title="4"><span class="bu">print</span>(u_v1)</a>
    343. 

  343. <a class="sourceLine" id="cb22-5" title="5"><span class="bu">print</span>(u_v2)</a>
    344. 

  344. <a class="sourceLine" id="cb22-6" title="6"></a>
    345. 

  345. <a class="sourceLine" id="cb22-7" title="7"><span class="cf">if</span> np.array_equal(u_v1, u_v2):</a>
    346. 

  346. <a class="sourceLine" id="cb22-8" title="8">    <span class="bu">print</span>(<span class="st">&#39;The vectors are dependent&#39;</span>)</a>
    347. 

  347. <a class="sourceLine" id="cb22-9" title="9"><span class="cf">else</span>:</a>
    348. 

  348. <a class="sourceLine" id="cb22-10" title="10">    <span class="bu">print</span>(<span class="st">&#39;The vectors are independent&#39;</span>)</a></code></pre></div>
    349. 

  349. <pre><code>[0.70710678 0.70710678]
    350. 

  350. [0.70710678 0.70710678]
    351. 

  351. The vectors are dependent</code></pre>
    352. 

  352. <h2 id="linear-independence">Linear Independence</h2>
    353. 

  353. <div class="sourceCode" id="cb24"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb24-1" title="1">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">2</span>])</a>
    354. 

  354. <a class="sourceLine" id="cb24-2" title="2">v2 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">2</span>])</a>
    355. 

  355. <a class="sourceLine" id="cb24-3" title="3"></a>
    356. 

  356. <a class="sourceLine" id="cb24-4" title="4"><span class="bu">print</span>(v1)</a>
    357. 

  357. <a class="sourceLine" id="cb24-5" title="5"><span class="bu">print</span>(v2)</a></code></pre></div>
    358. 

  358. <pre><code>[2 2]
    359. 

  359. [1 2]</code></pre>
    360. 

  360. <div class="sourceCode" id="cb26"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb26-1" title="1">u_v1 <span class="op">=</span> v1 <span class="op">/</span> np.linalg.norm(v1)</a>
    361. 

  361. <a class="sourceLine" id="cb26-2" title="2">u_v2 <span class="op">=</span> v2 <span class="op">/</span> np.linalg.norm(v2)</a>
    362. 

  362. <a class="sourceLine" id="cb26-3" title="3"></a>
    363. 

  363. <a class="sourceLine" id="cb26-4" title="4"><span class="bu">print</span>(u_v1)</a>
    364. 

  364. <a class="sourceLine" id="cb26-5" title="5"><span class="bu">print</span>(u_v2)</a>
    365. 

  365. <a class="sourceLine" id="cb26-6" title="6"></a>
    366. 

  366. <a class="sourceLine" id="cb26-7" title="7"><span class="cf">if</span> np.array_equal(u_v1, u_v2):</a>
    367. 

  367. <a class="sourceLine" id="cb26-8" title="8">    <span class="bu">print</span>(<span class="st">&#39;The vectors are dependent&#39;</span>)</a>
    368. 

  368. <a class="sourceLine" id="cb26-9" title="9"><span class="cf">else</span>:</a>
    369. 

  369. <a class="sourceLine" id="cb26-10" title="10">    <span class="bu">print</span>(<span class="st">&#39;The vectors are independent&#39;</span>)</a></code></pre></div>
    370. 

  370. <pre><code>[0.70710678 0.70710678]
    371. 

  371. [0.4472136  0.89442719]
    372. 

  372. The vectors are independent</code></pre>
    373. 

  373. <h2 id="general-check-for-independence">General Check For Independence</h2>
    374. 

  374. <p>$  = [1 ;; 2], ;;;  = [2 ;; 2] $</p>
    375. 

  375. <p>If there is <span class="math inline"><em>c</em><sub>1</sub> ≠ 0</span> for <span class="math inline"><em>c</em><sub>1</sub></span> being any real number</p>
    376. 

  376. <p>and / or some</p>
    377. 

  377. <p>$ c_2 0 $ for <span class="math inline"><em>c</em><sub>2</sub></span> being any real number</p>
    378. 

  378. <p>to satisfy the equation</p>
    379. 

  379. <p>$ 0 = c_1  + c_2  $</p>
    380. 

  380. <p>then, the vectors are dependent.</p>
    381. 

  381. <p>If $ c_1 $ and $ c_2 $ MUST BE zero to satisfy the equation above,</p>
    382. 

  382. <p>then the vectors are independent.</p>
    383. 

  383. <h2 id="visualization-of-above-principles">Visualization Of Above Principles</h2>
    384. 

  384. <h2 id="visualizing-linear-dependence">Visualizing Linear Dependence</h2>
    385. 

  385. <div class="sourceCode" id="cb28"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb28-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    386. 

  386. <a class="sourceLine" id="cb28-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    387. 

  387. <a class="sourceLine" id="cb28-3" title="3"></a>
    388. 

  388. <a class="sourceLine" id="cb28-4" title="4">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    389. 

  389. <a class="sourceLine" id="cb28-5" title="5">plt.xlim([<span class="dv">0</span>, <span class="dv">7</span>])<span class="op">;</span> plt.ylim([<span class="dv">0</span>, <span class="dv">7</span>])</a>
    390. 

  390. <a class="sourceLine" id="cb28-6" title="6"></a>
    391. 

  391. <a class="sourceLine" id="cb28-7" title="7">v1i <span class="op">=</span> <span class="dv">1</span><span class="op">;</span> v1j <span class="op">=</span> <span class="dv">1</span></a>
    392. 

  392. <a class="sourceLine" id="cb28-8" title="8">v2i <span class="op">=</span> <span class="dv">2</span><span class="op">;</span> v2j <span class="op">=</span> <span class="dv">2</span></a>
    393. 

  393. <a class="sourceLine" id="cb28-9" title="9"></a>
    394. 

  394. <a class="sourceLine" id="cb28-10" title="10">plt.arrow(<span class="dv">2</span>, <span class="dv">2</span>, v1i, v1j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    395. 

  395. <a class="sourceLine" id="cb28-11" title="11">plt.arrow(<span class="dv">2</span>, <span class="dv">2</span>, v2i, v2j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    396. 

  396. <a class="sourceLine" id="cb28-12" title="12"></a>
    397. 

  397. <a class="sourceLine" id="cb28-13" title="13">stuff <span class="op">=</span> <span class="vs">r&#39;Visualizing $\vec</span><span class="sc">{v_1}</span><span class="vs">$ and $\vec</span><span class="sc">{v_2}</span><span class="vs">$&#39;</span></a>
    398. 

  398. <a class="sourceLine" id="cb28-14" title="14">plt.title(stuff, color<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    399. 

  399. <a class="sourceLine" id="cb28-15" title="15">plt.show()<span class="op">;</span></a></code></pre></div>
    400. 

  400. <figure>
    401. 

  401. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_41_0.png" alt="png" /><figcaption>png</figcaption>
    402. 

  402. </figure>
    403. 

  403. <h2 id="visualizing-linear-independence">Visualizing Linear Independence</h2>
    404. 

  404. <div class="sourceCode" id="cb29"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb29-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    405. 

  405. <a class="sourceLine" id="cb29-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    406. 

  406. <a class="sourceLine" id="cb29-3" title="3"></a>
    407. 

  407. <a class="sourceLine" id="cb29-4" title="4">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    408. 

  408. <a class="sourceLine" id="cb29-5" title="5">plt.xlim([<span class="dv">0</span>, <span class="dv">7</span>])<span class="op">;</span> plt.ylim([<span class="dv">0</span>, <span class="dv">7</span>])</a>
    409. 

  409. <a class="sourceLine" id="cb29-6" title="6"></a>
    410. 

  410. <a class="sourceLine" id="cb29-7" title="7">v1i <span class="op">=</span> <span class="dv">1</span><span class="op">;</span> v1j <span class="op">=</span> <span class="dv">2</span></a>
    411. 

  411. <a class="sourceLine" id="cb29-8" title="8">v2i <span class="op">=</span> <span class="dv">2</span><span class="op">;</span> v2j <span class="op">=</span> <span class="dv">2</span></a>
    412. 

  412. <a class="sourceLine" id="cb29-9" title="9"></a>
    413. 

  413. <a class="sourceLine" id="cb29-10" title="10">plt.arrow(<span class="dv">2</span>, <span class="dv">2</span>, v1i, v1j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    414. 

  414. <a class="sourceLine" id="cb29-11" title="11">plt.arrow(<span class="dv">2</span>, <span class="dv">2</span>, v2i, v2j, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    415. 

  415. <a class="sourceLine" id="cb29-12" title="12"></a>
    416. 

  416. <a class="sourceLine" id="cb29-13" title="13">stuff <span class="op">=</span> <span class="vs">r&#39;Visualizing $\vec</span><span class="sc">{v_1}</span><span class="vs">$ and $\vec</span><span class="sc">{v_2}</span><span class="vs">$&#39;</span></a>
    417. 

  417. <a class="sourceLine" id="cb29-14" title="14">plt.title(stuff, color<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    418. 

  418. <a class="sourceLine" id="cb29-15" title="15">plt.show()<span class="op">;</span></a></code></pre></div>
    419. 

  419. <figure>
    420. 

  420. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_43_0.png" alt="png" /><figcaption>png</figcaption>
    421. 

  421. </figure>
    422. 

  422. <h2 id="takeaway">Takeaway</h2>
    423. 

  423. <p>With increased dimensionality comes new information.</p>
    424. 

  424. <h1 id="compact-review-linear-dependence-independence-visually-via-unit-vectorization">Compact Review<br>Linear Dependence &amp; Independence:<br>Visually &amp; Via Unit Vectorization</h1>
    425. 

  425. <div class="sourceCode" id="cb30"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb30-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    426. 

  426. <a class="sourceLine" id="cb30-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    427. 

  427. <a class="sourceLine" id="cb30-3" title="3"></a>
    428. 

  428. <a class="sourceLine" id="cb30-4" title="4">plt.figure(figsize<span class="op">=</span>(<span class="dv">7</span>, <span class="dv">4</span>))</a>
    429. 

  429. <a class="sourceLine" id="cb30-5" title="5">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    430. 

  430. <a class="sourceLine" id="cb30-6" title="6">plt.xlim([<span class="dv">0</span>, <span class="dv">7</span>])<span class="op">;</span> plt.ylim([<span class="dv">0</span>, <span class="dv">4</span>])</a>
    431. 

  431. <a class="sourceLine" id="cb30-7" title="7"></a>
    432. 

  432. <a class="sourceLine" id="cb30-8" title="8">v1 <span class="op">=</span> [<span class="dv">1</span>, <span class="dv">1</span>]<span class="op">;</span> v2 <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">2</span>]<span class="op">;</span> v3 <span class="op">=</span> [<span class="dv">1</span>, <span class="dv">2</span>]<span class="op">;</span> v4 <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">2</span>]</a>
    433. 

  433. <a class="sourceLine" id="cb30-9" title="9"></a>
    434. 

  434. <a class="sourceLine" id="cb30-10" title="10">u_v1 <span class="op">=</span> v1 <span class="op">/</span> np.linalg.norm(v1)<span class="op">;</span> u_v2 <span class="op">=</span> v2 <span class="op">/</span> np.linalg.norm(v2)</a>
    435. 

  435. <a class="sourceLine" id="cb30-11" title="11">u_v3 <span class="op">=</span> v3 <span class="op">/</span> np.linalg.norm(v3)<span class="op">;</span> u_v4 <span class="op">=</span> v4 <span class="op">/</span> np.linalg.norm(v4)</a>
    436. 

  436. <a class="sourceLine" id="cb30-12" title="12"></a>
    437. 

  437. <a class="sourceLine" id="cb30-13" title="13"><span class="bu">print</span>(<span class="ss">f&#39;Vectors 1 and 2 are independent: </span><span class="sc">{</span><span class="kw">not</span> <span class="sc">np.</span>array_equal(u_v1, u_v2)<span class="sc">}</span><span class="ss">&#39;</span>)</a>
    438. 

  438. <a class="sourceLine" id="cb30-14" title="14"><span class="bu">print</span>(<span class="ss">f&#39;Vectors 3 and 4 are independent: </span><span class="sc">{</span><span class="kw">not</span> <span class="sc">np.</span>array_equal(u_v3, u_v4)<span class="sc">}</span><span class="ch">\n</span><span class="ss">&#39;</span>)</a>
    439. 

  439. <a class="sourceLine" id="cb30-15" title="15"></a>
    440. 

  440. <a class="sourceLine" id="cb30-16" title="16">plt.arrow(<span class="dv">1</span>, <span class="dv">1</span>, v1[<span class="dv">0</span>], v1[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    441. 

  441. <a class="sourceLine" id="cb30-17" title="17">plt.arrow(<span class="dv">1</span>, <span class="dv">1</span>, v2[<span class="dv">0</span>], v2[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    442. 

  442. <a class="sourceLine" id="cb30-18" title="18">plt.arrow(<span class="dv">4</span>, <span class="dv">1</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    443. 

  443. <a class="sourceLine" id="cb30-19" title="19">plt.arrow(<span class="dv">4</span>, <span class="dv">1</span>, v4[<span class="dv">0</span>], v4[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    444. 

  444. <a class="sourceLine" id="cb30-20" title="20"></a>
    445. 

  445. <a class="sourceLine" id="cb30-21" title="21">stuff <span class="op">=</span> <span class="vs">r&#39;$\vec</span><span class="sc">{v_1}</span><span class="vs">$ and $\vec</span><span class="sc">{v_2}</span><span class="vs">$ are Dependent&#39;</span></a>
    446. 

  446. <a class="sourceLine" id="cb30-22" title="22">stuff <span class="op">+=</span> <span class="vs">r&#39; ... $\vec</span><span class="sc">{v_3}</span><span class="vs">$ and $\vec</span><span class="sc">{v_4}</span><span class="vs">$ are Independent&#39;</span></a>
    447. 

  447. <a class="sourceLine" id="cb30-23" title="23">plt.title(stuff, color<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    448. 

  448. <a class="sourceLine" id="cb30-24" title="24"></a>
    449. 

  449. <a class="sourceLine" id="cb30-25" title="25">plt.show()<span class="op">;</span></a></code></pre></div>
    450. 

  450. <pre><code>Vectors 1 and 2 are independent: False
    451. 

  451. Vectors 3 and 4 are independent: True</code></pre>
    452. 

  452. <figure>
    453. 

  453. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_46_1.png" alt="png" /><figcaption>png</figcaption>
    454. 

  454. </figure>
    455. 

  455. <h1 id="subspaces-and-the-basis-for-a-subspace">Subspaces And The Basis For A Subspace</h1>
    456. 

  456. <p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/subspace-basis/v/linear-subspaces">Subspaces And The Basis For A Subspace</a> * <a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/subspace-basis/v/linear-subspaces?modal=1">Linear Subspaces</a> * <a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/subspace-basis/v/linear-algebra-basis-of-a-subspace?modal=1">Basis Of A Subspace</a></p>
    457. 

  457. <p>Linear Algebra and the Data Sciences are subsets of STEM * Sometimes it takes learning several lectures or sessions or chapters of STEM material before the light turns on in your mind - be patient * It’s perfectly normal if or when you fall in love with this stuff to want to watch Sal’s lectures again, OR read a linear algebra textbook multiple times, and to do the same with multiple linear algebra textbooks OR materials. * Branches of math and science important to the data sciences: * Algebra * Linear Algebra * Trigonometry * Statistics (a science that uses the math of probability) * The Calculus</p>
    458. 

  458. <h2 id="how-to-determine-if-some-v-which-is-a-subset-of-ℝn-is-also-a-valid-subspace-of-ℝn">How To Determine IF Some <span class="math inline"><em>V</em></span>, Which Is A Subset Of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span> Is Also A Valid Subspace of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span></h2>
    459. 

  459. <p>Some <span class="math inline"><em>V</em></span> is a subset of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span>.</p>
    460. 

  460. <p>$  $ is a valid subspace of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span> IF the following three statements are true:</p>
    461. 

  461. <ol type="1">
    462. 

  462. <li>$  $ contains the zero vector for n dimensions … $  =
    463. 

  463. <p>$, <br> which means all vectors in <span class="math inline"><em>V⃗</em></span> are linearly independent of each other and no vectors in <span class="math inline"><em>V⃗</em></span> are linear combinations of each other.</p></li>
    464. 

  464. <li><p>If <span class="math inline"><em>x⃗</em></span> is in <span class="math inline"><em>V⃗</em></span>, then any <span class="math inline"><em>c</em><em>x⃗</em></span> is also in <span class="math inline"><em>V⃗</em></span> where <span class="math inline"><em>c</em></span> is any scalar value.</p></li>
    465. 

  465. <li><p>If $  $ is in $  $ and if $  $ is in $  $, then $  +  $ is in $  $</p></li>
    466. 

  466. </ol>
    467. 

  467. <p>This all means that if the set of vectors <span class="math inline"><em>V</em></span> is closed under multiplication and addition and contains the zero vector, it is a valid subspace of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span>.</p>
    468. 

  468. <h2 id="a-super-simple-example-of-a-valid-subspace-of-ℝ3">A Super Simple Example Of A Valid Subspace Of $ ℝ^3 $</h2>
    469. 

  469. <p><span class="math inline">$\vec{V} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$</span> is a subset of <span class="math inline"><em>ℝ</em><sup>3</sup></span>.</p>
    470. 

  470. <p>Is <span class="math inline"><em>V⃗</em></span> a valid subspace of <span class="math inline"><em>ℝ</em><sup>3</sup></span>?</p>
    471. 

  471. <ol type="1">
    472. 

  472. <li><p><span class="math inline"><em>V⃗</em></span> does contain the zero vector for 3 dimensions.</p></li>
    473. 

  473. <li><p>Any <span class="math inline"><em>c</em><em>V⃗</em></span> is also in <span class="math inline"><em>V⃗</em></span> where <span class="math inline"><em>c</em></span> is any scalar value.</p></li>
    474. 

  474. <li><p><span class="math inline"><em>V⃗</em> + <em>V⃗</em> = <em>V⃗</em></span></p></li>
    475. 

  475. </ol>
    476. 

  476. <p>Thus <span class="math inline"><em>V⃗</em></span> is a valid subspace of <span class="math inline"><em>ℝ</em><sup>3</sup></span>.</p>
    477. 

  477. <p>Even though this is trivially simple subspace, it is still valid subspace.</p>
    478. 

  478. <h2 id="a-simple-example-of-an-invalid-subspace-of-ℝ2">A Simple Example Of An Invalid Subspace Of <span class="math inline"><em>ℝ</em><sup>2</sup></span></h2>
    479. 

  479. <p><span class="math inline">$S = \begin{Bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} in\; ℝ^2 \mid x_1 \ge 0 \end{Bmatrix}$</span></p>
    480. 

  480. <p>Is <span class="math inline"><em>S</em></span> a valid subspace of <span class="math inline"><em>ℝ</em><sup>2</sup></span>?</p>
    481. 

  481. <ol type="1">
    482. 

  482. <li><p><span class="math inline"><em>S</em></span> does contain the zero vector for 2 dimensions.</p></li>
    483. 

  483. <li><p>NOT all <span class="math inline">$c \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$</span> are in <span class="math inline"><em>S</em></span> where <span class="math inline"><em>c</em></span> is any scalar value, so <span class="math inline"><em>S</em></span> is not closed under multiplication.</p></li>
    484. 

  484. <li><p><span class="math inline"><em>a⃗</em></span> is in <span class="math inline"><em>S</em></span> and <span class="math inline"><em>b⃗</em></span> is in <span class="math inline"><em>S</em></span> and <span class="math inline"><em>a⃗</em> + <em>b⃗</em></span> is in <span class="math inline"><em>S</em></span>, so <span class="math inline"><em>S</em></span> is closed under addition.</p></li>
    485. 

  485. </ol>
    486. 

  486. <p>Thus <span class="math inline"><em>S</em></span> is NOT a valid subspace of <span class="math inline"><em>ℝ</em><sup>2</sup></span>, because it is not closed for multiplication (rule 2).</p>
    487. 

  487. <p>Let’s illustrate this with Python, NumPy, and MatPlotLib</p>
    488. 

  488. <div class="sourceCode" id="cb32"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb32-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    489. 

  489. <a class="sourceLine" id="cb32-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    490. 

  490. <a class="sourceLine" id="cb32-3" title="3"></a>
    491. 

  491. <a class="sourceLine" id="cb32-4" title="4">plt.rc_context({</a>
    492. 

  492. <a class="sourceLine" id="cb32-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    493. 

  493. <a class="sourceLine" id="cb32-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    494. 

  494. <a class="sourceLine" id="cb32-7" title="7"></a>
    495. 

  495. <a class="sourceLine" id="cb32-8" title="8">x_list <span class="op">=</span> []</a>
    496. 

  496. <a class="sourceLine" id="cb32-9" title="9">finite_span <span class="op">=</span> <span class="dv">20</span></a>
    497. 

  497. <a class="sourceLine" id="cb32-10" title="10">lim <span class="op">=</span> <span class="dv">30</span></a>
    498. 

  498. <a class="sourceLine" id="cb32-11" title="11">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    499. 

  499. <a class="sourceLine" id="cb32-12" title="12"><span class="cf">for</span> x1 <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">0</span>, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    500. 

  500. <a class="sourceLine" id="cb32-13" title="13">    <span class="cf">for</span> x2 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    501. 

  501. <a class="sourceLine" id="cb32-14" title="14">        x <span class="op">=</span> np.array([x1, x2])</a>
    502. 

  502. <a class="sourceLine" id="cb32-15" title="15">        x_list.append(x)</a>
    503. 

  503. <a class="sourceLine" id="cb32-16" title="16">        plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    504. 

  504. <a class="sourceLine" id="cb32-17" title="17">        plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    505. 

  505. <a class="sourceLine" id="cb32-18" title="18">                  width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    506. 

  506. <a class="sourceLine" id="cb32-19" title="19"></a>
    507. 

  507. <a class="sourceLine" id="cb32-20" title="20">plt.xlim([<span class="op">-</span>lim, lim])</a>
    508. 

  508. <a class="sourceLine" id="cb32-21" title="21">plt.ylim([<span class="op">-</span>lim, lim])</a>
    509. 

  509. <a class="sourceLine" id="cb32-22" title="22">plt.show()<span class="op">;</span></a></code></pre></div>
    510. 

  510. <figure>
    511. 

  511. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_53_0.png" alt="png" /><figcaption>png</figcaption>
    512. 

  512. </figure>
    513. 

  513. <div class="sourceCode" id="cb33"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb33-1" title="1">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    514. 

  514. <a class="sourceLine" id="cb33-2" title="2"><span class="cf">for</span> x <span class="kw">in</span> x_list:</a>
    515. 

  515. <a class="sourceLine" id="cb33-3" title="3">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    516. 

  516. <a class="sourceLine" id="cb33-4" title="4">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    517. 

  517. <a class="sourceLine" id="cb33-5" title="5">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    518. 

  518. <a class="sourceLine" id="cb33-6" title="6"></a>
    519. 

  519. <a class="sourceLine" id="cb33-7" title="7">plt.xlim([<span class="op">-</span>lim, lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    520. 

  520. <a class="sourceLine" id="cb33-8" title="8">plt.show()<span class="op">;</span></a></code></pre></div>
    521. 

  521. <figure>
    522. 

  522. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_54_0.png" alt="png" /><figcaption>png</figcaption>
    523. 

  523. </figure>
    524. 

  524. <div class="sourceCode" id="cb34"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb34-1" title="1"><span class="co"># is [0 0] in x_list, and x_list in a python list of numpy vectors / arrays</span></a>
    525. 

  525. <a class="sourceLine" id="cb34-2" title="2"><span class="co"># We want a python list of python arrays so we can use the &quot;in&quot; logic</span></a>
    526. 

  526. <a class="sourceLine" id="cb34-3" title="3">py_list_of_py_lists <span class="op">=</span> [x.tolist() <span class="cf">for</span> x <span class="kw">in</span> x_list]</a>
    527. 

  527. <a class="sourceLine" id="cb34-4" title="4"><span class="bu">print</span>([<span class="dv">0</span>, <span class="dv">0</span>] <span class="kw">in</span> py_list_of_py_lists)</a></code></pre></div>
    528. 

  528. <pre><code>True</code></pre>
    529. 

  529. <div class="sourceCode" id="cb36"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb36-1" title="1"><span class="im">import</span> random</a>
    530. 

  530. <a class="sourceLine" id="cb36-2" title="2"></a>
    531. 

  531. <a class="sourceLine" id="cb36-3" title="3">list_1 <span class="op">=</span> random.sample(x_list, <span class="dv">1</span>)[<span class="dv">0</span>]</a>
    532. 

  532. <a class="sourceLine" id="cb36-4" title="4">list_2 <span class="op">=</span> random.sample(x_list, <span class="dv">1</span>)[<span class="dv">0</span>]</a>
    533. 

  533. <a class="sourceLine" id="cb36-5" title="5"><span class="bu">print</span>(list_1, list_2)</a>
    534. 

  534. <a class="sourceLine" id="cb36-6" title="6">sum_of_any_two_lists <span class="op">=</span> list_1 <span class="op">+</span> list_2</a>
    535. 

  535. <a class="sourceLine" id="cb36-7" title="7"><span class="bu">print</span>(sum_of_any_two_lists)</a>
    536. 

  536. <a class="sourceLine" id="cb36-8" title="8"><span class="bu">print</span>(sum_of_any_two_lists.tolist() <span class="kw">in</span> py_list_of_py_lists)</a></code></pre></div>
    537. 

  537. <pre><code>[10 13] [  5 -16]
    538. 

  538. [15 -3]
    539. 

  539. True</code></pre>
    540. 

  540. <div class="sourceCode" id="cb38"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb38-1" title="1">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    541. 

  541. <a class="sourceLine" id="cb38-2" title="2"><span class="cf">for</span> x <span class="kw">in</span> x_list:</a>
    542. 

  542. <a class="sourceLine" id="cb38-3" title="3">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    543. 

  543. <a class="sourceLine" id="cb38-4" title="4">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    544. 

  544. <a class="sourceLine" id="cb38-5" title="5">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    545. 

  545. <a class="sourceLine" id="cb38-6" title="6"></a>
    546. 

  546. <a class="sourceLine" id="cb38-7" title="7">bool_list <span class="op">=</span> []</a>
    547. 

  547. <a class="sourceLine" id="cb38-8" title="8">x <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">1</span>])</a>
    548. 

  548. <a class="sourceLine" id="cb38-9" title="9"><span class="cf">for</span> c <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span><span class="dv">10</span>, <span class="dv">10</span><span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    549. 

  549. <a class="sourceLine" id="cb38-10" title="10">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    550. 

  550. <a class="sourceLine" id="cb38-11" title="11">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, c<span class="op">*</span>x[<span class="dv">0</span>], c<span class="op">*</span>x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    551. 

  551. <a class="sourceLine" id="cb38-12" title="12">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    552. 

  552. <a class="sourceLine" id="cb38-13" title="13">    curr_vec <span class="op">=</span> c<span class="op">*</span>x</a>
    553. 

  553. <a class="sourceLine" id="cb38-14" title="14">    bool_list.append(curr_vec.tolist() <span class="kw">in</span> py_list_of_py_lists)</a>
    554. 

  554. <a class="sourceLine" id="cb38-15" title="15"></a>
    555. 

  555. <a class="sourceLine" id="cb38-16" title="16"><span class="bu">print</span>(bool_list)</a>
    556. 

  556. <a class="sourceLine" id="cb38-17" title="17">plt.xlim([<span class="op">-</span>lim, lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    557. 

  557. <a class="sourceLine" id="cb38-18" title="18">plt.show()<span class="op">;</span></a></code></pre></div>
    558. 

  558. <pre><code>[False, False, False, False, False, False, False, False, False, False, True, True, True, True, True, True, True, True, True, True, True]</code></pre>
    559. 

  559. <figure>
    560. 

  560. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_57_1.png" alt="png" /><figcaption>png</figcaption>
    561. 

  561. </figure>
    562. 

  562. <h2 id="a-very-general-example-for-a-span-in-ℝn">A VERY General Example For A Span In <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span></h2>
    563. 

  563. <p>Note, that in this example, we do not even know the number of dimensions <span class="math inline"><em>n</em></span>.</p>
    564. 

  564. <p>Is $ U = Span(, , )$ a valid subspace of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span><br> &gt; where each of the vectors <span class="math inline">$\vec{V_1}, \vec{V_2}, \vec{V_3}$</span> have dimension <span class="math inline"><em>n</em></span>?</p>
    565. 

  565. <ol type="1">
    566. 

  566. <li><p>$ 0  + 0  + 0  = $</p></li>
    567. 

  567. <li><p>$  = c_1  + c_2  + c_3 $ <br> $ a  = a c_1  + a c_2  + a c_3 $ <br> $ a  = c_4  + c_5  + c_6 $ <br> all of these are in the <span class="math inline"><em>S</em><em>p</em><em>a</em><em>n</em></span> of <span class="math inline"><em>U</em></span>, which is our subspace, so U is closed under multiplication.</p></li>
    568. 

  568. <li><p>Let’s define another vector <br> <span class="math inline">$\vec{y} = d_1 \vec{V_1} + d_2 \vec{V_2} + d_3 \vec{V_3}$</span><br> <span class="math inline">$\vec{x} + \vec{y} = (c_1 + d_1) \vec{V_1} + (c_2 + d_2) \vec{V_2} + (c_3 + d_3) \vec{V_3}$</span><br> but <span class="math inline"><em>x⃗</em> + <em>y⃗</em></span> is just a linear combination of <span class="math inline">$\vec{V_1}, \vec{V_2}, \vec{V_3}$</span>, which would be in the span of <span class="math inline"><em>U</em></span>, so U is closed under addition.</p></li>
    569. 

  569. </ol>
    570. 

  570. <p>Thus, <span class="math inline"><em>U</em></span> is a valid subspace of <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span></p>
    571. 

  571. <h2 id="a-simple-example-of-a-valid-subspace-in-ℝ2">A Simple Example Of A Valid Subspace in <span class="math inline"><em>ℝ</em><sup>2</sup></span></h2>
    572. 

  572. <p>Is <span class="math inline">$U = Span \begin{pmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \end{pmatrix}$</span> a valid subspace of <span class="math inline"><em>ℝ</em><sup>2</sup></span> ?</p>
    573. 

  573. <ol type="1">
    574. 

  574. <li>First, $ 0
    575. 

  575. =
    576. 

  576. <p>$, so this rule is valid.</p></li>
    577. 

  577. <li>Any $  = c
    578. 

  578. =
    579. 

  579. $ <br> and all $
    580. 

  580. <p>$ would be in <span class="math inline"><em>U</em></span>, so U is closed under multiplication.</p></li>
    581. 

  581. <li><p>Some other vector <span class="math inline">$\vec{y} = d \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} d \\ d \end{bmatrix}$</span> would be in <span class="math inline"><em>U</em></span> by the previous rule, <br> and <span class="math inline">$\vec{x} + \vec{y} = \begin{bmatrix} c+d \\ c+d \end{bmatrix}$</span>,<br> which is just a linear combination of <span class="math inline">$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$</span> and would thus be in <span class="math inline"><em>U</em></span>, so <span class="math inline"><em>U</em></span> is closed under addition.</p></li>
    582. 

  582. </ol>
    583. 

  583. <p>Thus, <span class="math inline">$U = Span \begin{pmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \end{pmatrix}$</span> IS a valid subspace of <span class="math inline"><em>ℝ</em><sup>2</sup></span></p>
    584. 

  584. <p>Let’s illustrate this with Python, NumPy, and MatPlotLib</p>
    585. 

  585. <div class="sourceCode" id="cb40"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb40-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    586. 

  586. <a class="sourceLine" id="cb40-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    587. 

  587. <a class="sourceLine" id="cb40-3" title="3"></a>
    588. 

  588. <a class="sourceLine" id="cb40-4" title="4">plt.rc_context({</a>
    589. 

  589. <a class="sourceLine" id="cb40-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    590. 

  590. <a class="sourceLine" id="cb40-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    591. 

  591. <a class="sourceLine" id="cb40-7" title="7"></a>
    592. 

  592. <a class="sourceLine" id="cb40-8" title="8">x_list <span class="op">=</span> []</a>
    593. 

  593. <a class="sourceLine" id="cb40-9" title="9">finite_span <span class="op">=</span> <span class="dv">20</span></a>
    594. 

  594. <a class="sourceLine" id="cb40-10" title="10">lim <span class="op">=</span> <span class="dv">30</span></a>
    595. 

  595. <a class="sourceLine" id="cb40-11" title="11">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    596. 

  596. <a class="sourceLine" id="cb40-12" title="12">x <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">1</span>])</a>
    597. 

  597. <a class="sourceLine" id="cb40-13" title="13"><span class="cf">for</span> c <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    598. 

  598. <a class="sourceLine" id="cb40-14" title="14">    x_list.append(c <span class="op">*</span> x)</a>
    599. 

  599. <a class="sourceLine" id="cb40-15" title="15">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    600. 

  600. <a class="sourceLine" id="cb40-16" title="16">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, c<span class="op">*</span>x[<span class="dv">0</span>], c<span class="op">*</span>x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    601. 

  601. <a class="sourceLine" id="cb40-17" title="17">                width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    602. 

  602. <a class="sourceLine" id="cb40-18" title="18"></a>
    603. 

  603. <a class="sourceLine" id="cb40-19" title="19">plt.xlim([<span class="op">-</span>lim, lim])</a>
    604. 

  604. <a class="sourceLine" id="cb40-20" title="20">plt.ylim([<span class="op">-</span>lim, lim])</a>
    605. 

  605. <a class="sourceLine" id="cb40-21" title="21">plt.show()<span class="op">;</span></a></code></pre></div>
    606. 

  606. <figure>
    607. 

  607. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_60_0.png" alt="png" /><figcaption>png</figcaption>
    608. 

  608. </figure>
    609. 

  609. <div class="sourceCode" id="cb41"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb41-1" title="1">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    610. 

  610. <a class="sourceLine" id="cb41-2" title="2"><span class="cf">for</span> x <span class="kw">in</span> x_list:</a>
    611. 

  611. <a class="sourceLine" id="cb41-3" title="3">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    612. 

  612. <a class="sourceLine" id="cb41-4" title="4">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    613. 

  613. <a class="sourceLine" id="cb41-5" title="5">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    614. 

  614. <a class="sourceLine" id="cb41-6" title="6"></a>
    615. 

  615. <a class="sourceLine" id="cb41-7" title="7">plt.xlim([<span class="op">-</span>lim, lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    616. 

  616. <a class="sourceLine" id="cb41-8" title="8">plt.show()<span class="op">;</span></a></code></pre></div>
    617. 

  617. <figure>
    618. 

  618. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_61_0.png" alt="png" /><figcaption>png</figcaption>
    619. 

  619. </figure>
    620. 

  620. <div class="sourceCode" id="cb42"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb42-1" title="1"><span class="co"># is [0 0] in x_list, and x_list in a python list of numpy vectors / arrays</span></a>
    621. 

  621. <a class="sourceLine" id="cb42-2" title="2"><span class="co"># We want a python list of python arrays so we can use the &quot;in&quot; logic</span></a>
    622. 

  622. <a class="sourceLine" id="cb42-3" title="3">py_list_of_py_lists <span class="op">=</span> [x.tolist() <span class="cf">for</span> x <span class="kw">in</span> x_list]</a>
    623. 

  623. <a class="sourceLine" id="cb42-4" title="4"><span class="bu">print</span>([<span class="dv">0</span>, <span class="dv">0</span>] <span class="kw">in</span> py_list_of_py_lists)</a></code></pre></div>
    624. 

  624. <pre><code>True</code></pre>
    625. 

  625. <div class="sourceCode" id="cb44"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb44-1" title="1"><span class="im">import</span> random</a>
    626. 

  626. <a class="sourceLine" id="cb44-2" title="2"></a>
    627. 

  627. <a class="sourceLine" id="cb44-3" title="3">list_1 <span class="op">=</span> random.sample(x_list, <span class="dv">1</span>)[<span class="dv">0</span>]</a>
    628. 

  628. <a class="sourceLine" id="cb44-4" title="4">list_2 <span class="op">=</span> random.sample(x_list, <span class="dv">1</span>)[<span class="dv">0</span>]</a>
    629. 

  629. <a class="sourceLine" id="cb44-5" title="5"><span class="bu">print</span>(list_1, list_2)</a>
    630. 

  630. <a class="sourceLine" id="cb44-6" title="6">sum_of_any_two_lists <span class="op">=</span> list_1 <span class="op">+</span> list_2</a>
    631. 

  631. <a class="sourceLine" id="cb44-7" title="7"><span class="bu">print</span>(sum_of_any_two_lists)</a>
    632. 

  632. <a class="sourceLine" id="cb44-8" title="8"><span class="bu">print</span>(sum_of_any_two_lists.tolist() <span class="kw">in</span> py_list_of_py_lists)</a></code></pre></div>
    633. 

  633. <pre><code>[-4 -4] [14 14]
    634. 

  634. [10 10]
    635. 

  635. True</code></pre>
    636. 

  636. <div class="sourceCode" id="cb46"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb46-1" title="1">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    637. 

  637. <a class="sourceLine" id="cb46-2" title="2"><span class="cf">for</span> x <span class="kw">in</span> x_list:</a>
    638. 

  638. <a class="sourceLine" id="cb46-3" title="3">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    639. 

  639. <a class="sourceLine" id="cb46-4" title="4">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    640. 

  640. <a class="sourceLine" id="cb46-5" title="5">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    641. 

  641. <a class="sourceLine" id="cb46-6" title="6"></a>
    642. 

  642. <a class="sourceLine" id="cb46-7" title="7">bool_list <span class="op">=</span> []</a>
    643. 

  643. <a class="sourceLine" id="cb46-8" title="8">x <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">1</span>])</a>
    644. 

  644. <a class="sourceLine" id="cb46-9" title="9"><span class="cf">for</span> c <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span><span class="dv">10</span>, <span class="dv">10</span><span class="op">+</span><span class="dv">1</span>):</a>
    645. 

  645. <a class="sourceLine" id="cb46-10" title="10">    plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    646. 

  646. <a class="sourceLine" id="cb46-11" title="11">    plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, c<span class="op">*</span>x[<span class="dv">0</span>], c<span class="op">*</span>x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    647. 

  647. <a class="sourceLine" id="cb46-12" title="12">              width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;red&#39;</span>)</a>
    648. 

  648. <a class="sourceLine" id="cb46-13" title="13">    curr_vec <span class="op">=</span> c<span class="op">*</span>x</a>
    649. 

  649. <a class="sourceLine" id="cb46-14" title="14">    bool_list.append(curr_vec.tolist() <span class="kw">in</span> py_list_of_py_lists)</a>
    650. 

  650. <a class="sourceLine" id="cb46-15" title="15"></a>
    651. 

  651. <a class="sourceLine" id="cb46-16" title="16"><span class="bu">print</span>(bool_list)</a>
    652. 

  652. <a class="sourceLine" id="cb46-17" title="17">plt.xlim([<span class="op">-</span>lim, lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    653. 

  653. <a class="sourceLine" id="cb46-18" title="18">plt.show()<span class="op">;</span></a></code></pre></div>
    654. 

  654. <pre><code>[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]</code></pre>
    655. 

  655. <figure>
    656. 

  656. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_64_1.png" alt="png" /><figcaption>png</figcaption>
    657. 

  657. </figure>
    658. 

  658. <h2 id="the-basis-for-a-subspace-is-a-set-of-linearly-independent-vectors">The Basis For A Subspace Is A Set Of Linearly Independent Vectors</h2>
    659. 

  659. <p>Let’s say that $ V = Span(, , , )$ is a subspace of some <span class="math inline"><em>ℝ</em><sup><em>n</em></sup></span><br> &gt; where each of the vectors <span class="math inline">$\vec{V_1}, \vec{V_2}, \dots, \vec{V_m}$</span> are linearly independent <span class="math inline">∴</span> no vectors in <span class="math inline"><em>V</em></span> are linear combinations of other vectors in <span class="math inline"><em>V</em></span>.</p>
    660. 

  660. <p>Then, if <span class="math inline">$S = {\vec{V_1}, \vec{V_2}, \dots, \vec{V_m}}$</span>, <span class="math inline"><em>S</em></span> is a <strong><em>basis</em></strong> for <span class="math inline"><em>V</em></span>.</p>
    661. 

  661. <p>However, if <span class="math inline">$T = {\vec{V_1}, \vec{V_2}, \dots, \vec{V_m}, \vec{V_s}}$</span>, and <span class="math inline">$\vec{V_s} = \vec{V_1} + \vec{V_2}$</span>, we can say that <span class="math inline"><em>S</em><em>p</em><em>a</em><em>n</em>(<em>T</em>) = <em>V</em></span>, but <span class="math inline"><em>T</em></span> is linearly dependent, and thus <span class="math inline"><em>T</em></span> cannot be a basis for <span class="math inline"><em>V</em></span>. Thus, <span class="math inline">$\vec{V_s}$</span> is redundant.</p>
    662. 

  662. <p>Thus a basis is the minimum set of vectors that spans the subspace.</p>
    663. 

  663. <h2 id="an-example-of-basis-of-a-subspace">An Example Of Basis Of A Subspace</h2>
    664. 

  664. <p><span class="math inline">$S = \begin{Bmatrix}  \begin{bmatrix} 2 \\ 3 \end{bmatrix}  \begin{bmatrix} 7 \\ 0 \end{bmatrix} \end{Bmatrix}$</span>. Is the Span of <span class="math inline"><em>S</em></span> a basis for <span class="math inline"><em>ℝ</em><sup>2</sup></span> ? Yes!</p>
    665. 

  665. <p>Please note that <span class="math inline"><em>S</em></span> is not the only basis for <span class="math inline"><em>ℝ</em><sup>2</sup></span>.</p>
    666. 

  666. <p>See Sal’s lecture for how to prove it algebraically. But note too that the two vectors are independent. One of them cannot be defined by terms of the other by scaling one to match the other.</p>
    667. 

  667. <p>Apply the rules that we covered above.</p>
    668. 

  668. <p>Let’s illustrate below with some code reused and modified from above. Remember, we have to <strong><em>imagine</em></strong> how the space with <span class="math inline"><em>c</em><sub><em>i</em></sub> ∈ <em>ℝ</em></span>. We are only using a finite set of <span class="math inline"><em>c</em><sub><em>i</em></sub></span>.</p>
    669. 

  669. <div class="sourceCode" id="cb48"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb48-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    670. 

  670. <a class="sourceLine" id="cb48-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    671. 

  671. <a class="sourceLine" id="cb48-3" title="3"></a>
    672. 

  672. <a class="sourceLine" id="cb48-4" title="4">v1 <span class="op">=</span> np.array([<span class="dv">2</span>, <span class="dv">3</span>])<span class="op">;</span> v2 <span class="op">=</span> np.array([<span class="dv">7</span>, <span class="dv">0</span>])</a>
    673. 

  673. <a class="sourceLine" id="cb48-5" title="5"></a>
    674. 

  674. <a class="sourceLine" id="cb48-6" title="6">finite_span <span class="op">=</span> <span class="dv">20</span></a>
    675. 

  675. <a class="sourceLine" id="cb48-7" title="7">lim <span class="op">=</span> <span class="dv">70</span></a>
    676. 

  676. <a class="sourceLine" id="cb48-8" title="8">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    677. 

  677. <a class="sourceLine" id="cb48-9" title="9"><span class="cf">for</span> c1 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    678. 

  678. <a class="sourceLine" id="cb48-10" title="10">    <span class="cf">for</span> c2 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    679. 

  679. <a class="sourceLine" id="cb48-11" title="11">        v3 <span class="op">=</span> c1<span class="op">*</span>v1 <span class="op">+</span> c2<span class="op">*</span>v2</a>
    680. 

  680. <a class="sourceLine" id="cb48-12" title="12">        plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    681. 

  681. <a class="sourceLine" id="cb48-13" title="13">        plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    682. 

  682. <a class="sourceLine" id="cb48-14" title="14">                  width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    683. 

  683. <a class="sourceLine" id="cb48-15" title="15"></a>
    684. 

  684. <a class="sourceLine" id="cb48-16" title="16">plt.xlim([<span class="op">-</span><span class="dv">3</span><span class="op">*</span>lim, <span class="dv">3</span><span class="op">*</span>lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    685. 

  685. <a class="sourceLine" id="cb48-17" title="17">plt.show()<span class="op">;</span></a></code></pre></div>
    686. 

  686. <figure>
    687. 

  687. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_67_0.png" alt="png" /><figcaption>png</figcaption>
    688. 

  688. </figure>
    689. 

  689. <h2 id="another-example-of-a-basis-of-a-subspace-ℝ2">Another Example Of A Basis Of A Subspace <span class="math inline"><em>ℝ</em><sup>2</sup></span></h2>
    690. 

  690. <p><span class="math inline">$T = \begin{Bmatrix}  \begin{bmatrix} 1 \\ 0 \end{bmatrix}  \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{Bmatrix}$</span>. Is the Span of <span class="math inline"><em>T</em></span> a basis for <span class="math inline"><em>ℝ</em><sup>2</sup></span> ? Yes.</p>
    691. 

  691. <p>Apply the rules that we covered above.</p>
    692. 

  692. <p>See Sal’s lecture for how to prove it algebraically.</p>
    693. 

  693. <p>Let’s illustrate again below with some code reused and modified from above.</p>
    694. 

  694. <div class="sourceCode" id="cb49"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb49-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    695. 

  695. <a class="sourceLine" id="cb49-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    696. 

  696. <a class="sourceLine" id="cb49-3" title="3"></a>
    697. 

  697. <a class="sourceLine" id="cb49-4" title="4">plt.rc_context({</a>
    698. 

  698. <a class="sourceLine" id="cb49-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    699. 

  699. <a class="sourceLine" id="cb49-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    700. 

  700. <a class="sourceLine" id="cb49-7" title="7"></a>
    701. 

  701. <a class="sourceLine" id="cb49-8" title="8">v1 <span class="op">=</span> np.array([<span class="dv">1</span>, <span class="dv">0</span>])<span class="op">;</span> v2 <span class="op">=</span> np.array([<span class="dv">0</span>, <span class="dv">1</span>])</a>
    702. 

  702. <a class="sourceLine" id="cb49-9" title="9"><span class="bu">print</span>(v1)</a>
    703. 

  703. <a class="sourceLine" id="cb49-10" title="10"><span class="bu">print</span>(v2)</a>
    704. 

  704. <a class="sourceLine" id="cb49-11" title="11"></a>
    705. 

  705. <a class="sourceLine" id="cb49-12" title="12">finite_span <span class="op">=</span> <span class="dv">20</span></a>
    706. 

  706. <a class="sourceLine" id="cb49-13" title="13">lim <span class="op">=</span> <span class="dv">30</span></a>
    707. 

  707. <a class="sourceLine" id="cb49-14" title="14">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>,<span class="dv">6</span>))</a>
    708. 

  708. <a class="sourceLine" id="cb49-15" title="15"><span class="cf">for</span> c1 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    709. 

  709. <a class="sourceLine" id="cb49-16" title="16">    <span class="cf">for</span> c2 <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>, <span class="dv">1</span>):</a>
    710. 

  710. <a class="sourceLine" id="cb49-17" title="17">        v3 <span class="op">=</span> c1<span class="op">*</span>v1 <span class="op">+</span> c2<span class="op">*</span>v2</a>
    711. 

  711. <a class="sourceLine" id="cb49-18" title="18">        plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    712. 

  712. <a class="sourceLine" id="cb49-19" title="19">        plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, v3[<span class="dv">0</span>], v3[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.5</span>,</a>
    713. 

  713. <a class="sourceLine" id="cb49-20" title="20">                  width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;green&#39;</span>)</a>
    714. 

  714. <a class="sourceLine" id="cb49-21" title="21"></a>
    715. 

  715. <a class="sourceLine" id="cb49-22" title="22">plt.xlim([<span class="op">-</span>lim, lim])<span class="op">;</span> plt.ylim([<span class="op">-</span>lim, lim])</a>
    716. 

  716. <a class="sourceLine" id="cb49-23" title="23">plt.show()<span class="op">;</span></a></code></pre></div>
    717. 

  717. <pre><code>[1 0]
    718. 

  718. [0 1]</code></pre>
    719. 

  719. <figure>
    720. 

  720. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_69_1.png" alt="png" /><figcaption>png</figcaption>
    721. 

  721. </figure>
    722. 

  722. <h2 id="another-example-of-a-basis-of-a-subspace-ℝ3">Another Example Of A Basis Of A Subspace <span class="math inline"><em>ℝ</em><sup>3</sup></span></h2>
    723. 

  723. <p><span class="math inline">$T = \begin{Bmatrix}  \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}  \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{Bmatrix}$</span>. Is the Span of <span class="math inline"><em>T</em></span> a basis for <span class="math inline"><em>ℝ</em><sup>3</sup></span> ? Yes. <br><br> <span class="math inline">$U = \begin{Bmatrix}  \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}  \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}  \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \end{Bmatrix}$</span>. Is the Span of <span class="math inline"><em>U</em></span> a basis for <span class="math inline"><em>ℝ</em><sup>3</sup></span> ? NO! <br><br> Why? The sum of the first <span class="math inline">3</span> vectors equals the <span class="math inline">4<sup><em>t</em><em>h</em></sup></span> vector.</p>
    724. 

  724. <h1 id="vector-dot-and-cross-products">Vector Dot And Cross Products</h1>
    725. 

  725. <p><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length">Vector Dot And Cross Products</a></p>
    726. 

  726. <ul>
    727. 

  727. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length?modal=1">Vector Dot Product And Vector Length</a></li>
    728. 

  728. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/proving-vector-dot-product-properties?modal=1">Proving Vector Dot Product Properties</a></li>
    729. 

  729. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/proof-of-the-cauchy-schwarz-inequality?modal=1">Proof Of The Cauchy-Schwarz Inequality</a></li>
    730. 

  730. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/linear-algebra-vector-triangle-inequality?modal=1">Vector Triangle Inequality</a></li>
    731. 

  731. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/defining-the-angle-between-vectors?modal=1">Defining The Angle Between Vectors</a></li>
    732. 

  732. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/defining-a-plane-in-r3-with-a-point-and-normal-vector?modal=1">Defining A Plane In R3 With A Point And Normal Vector</a></li>
    733. 

  733. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/linear-algebra-cross-product-introduction?modal=1">Cross Product Introduction</a></li>
    734. 

  734. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/proof-relationship-between-cross-product-and-sin-of-angle?modal=1">Proof: Relationship Between Cross Product And Sin Of Angle</a></li>
    735. 

  735. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/dot-and-cross-product-comparison-intuition?modal=1">Dot And Cross Product Comparison/Intuition</a></li>
    736. 

  736. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-triple-product-expansion-very-optional?modal=1">Vector Triple Product Expansion (Very Optional)</a></li>
    737. 

  737. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/normal-vector-from-plane-equation?modal=1">Normal Vector From Plane Equation</a></li>
    738. 

  738. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/point-distance-to-plane?modal=1">Point Distance To Plane</a></li>
    739. 

  739. <li><a href="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/distance-between-planes?modal=1">Distance Between Planes</a></li>
    740. 

  740. </ul>
    741. 

  741. <h2 id="vector-dot-product-and-vector-length">Vector Dot Product And Vector Length</h2>
    742. 

  742. <p><br> The dot product of two vectors is the sum of the products of their individual elements / components <br><br> <br /><span class="math display">$$ \vec{x} \cdot \vec{y} = 
    743. 

  743. \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} 
    744. 

  744. \cdot 
    745. 

  745. \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} 
    746. 

  746. = x_1 y_1 + x_2 y_2 + \dots + x_n y_n $$</span><br /> <br><br> Also, the length of a vector $  $ is the square root of the dot product of that vector onto itself …</p>
    747. 

  747. <p><br /><span class="math display">$$ \lVert \vec{a} \rVert = \sqrt{ \vec{a} \cdot \vec{a} } = 
    748. 

  748. \sqrt{ a_1 a_1 + a_2 a_2 + \dots + a_n a_n } $$</span><br /> <br><br> <strong>Note that dot products of vectors yield scalars</strong></p>
    749. 

  749. <div class="sourceCode" id="cb51"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb51-1" title="1">v1 <span class="op">=</span> [<span class="dv">1</span>, <span class="dv">1</span>]<span class="op">;</span> v2 <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">2</span>]</a>
    750. 

  750. <a class="sourceLine" id="cb51-2" title="2"></a>
    751. 

  751. <a class="sourceLine" id="cb51-3" title="3">v1_dot_v2 <span class="op">=</span> v1[<span class="dv">0</span>] <span class="op">*</span> v2[<span class="dv">0</span>] <span class="op">+</span> v1[<span class="dv">1</span>] <span class="op">*</span> v2[<span class="dv">1</span>]</a>
    752. 

  752. <a class="sourceLine" id="cb51-4" title="4"><span class="bu">print</span>(<span class="ss">f&#39;The dot product of v1 and v2 = </span><span class="sc">{</span>v1_dot_v2<span class="sc">}</span><span class="ss">&#39;</span>)</a></code></pre></div>
    753. 

  753. <pre><code>The dot product of v1 and v2 = 4</code></pre>
    754. 

  754. <div class="sourceCode" id="cb53"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb53-1" title="1">v1 <span class="op">=</span> [<span class="dv">1</span>, <span class="dv">1</span>]<span class="op">;</span> v2 <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">2</span>]</a>
    755. 

  755. <a class="sourceLine" id="cb53-2" title="2"></a>
    756. 

  756. <a class="sourceLine" id="cb53-3" title="3"><span class="kw">def</span> dot_product(va, vb):</a>
    757. 

  757. <a class="sourceLine" id="cb53-4" title="4">    <span class="cf">assert</span> <span class="bu">len</span>(va) <span class="op">==</span> <span class="bu">len</span>(vb), <span class="st">&quot;Vectors are not the same dimension&quot;</span></a>
    758. 

  758. <a class="sourceLine" id="cb53-5" title="5">    va_dot_vb <span class="op">=</span> <span class="dv">0</span></a>
    759. 

  759. <a class="sourceLine" id="cb53-6" title="6">    <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(va)):</a>
    760. 

  760. <a class="sourceLine" id="cb53-7" title="7">        va_dot_vb <span class="op">+=</span> va[i] <span class="op">*</span> vb[i]</a>
    761. 

  761. <a class="sourceLine" id="cb53-8" title="8"></a>
    762. 

  762. <a class="sourceLine" id="cb53-9" title="9">    <span class="cf">return</span> va_dot_vb</a>
    763. 

  763. <a class="sourceLine" id="cb53-10" title="10"></a>
    764. 

  764. <a class="sourceLine" id="cb53-11" title="11"><span class="bu">print</span>(<span class="ss">f&#39;The dot product of v1 and v2 = </span><span class="sc">{</span>dot_product(v1, v2)<span class="sc">}</span><span class="ss">&#39;</span>)</a></code></pre></div>
    765. 

  765. <pre><code>The dot product of v1 and v2 = 4</code></pre>
    766. 

  766. <div class="sourceCode" id="cb55"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb55-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    767. 

  767. <a class="sourceLine" id="cb55-2" title="2"></a>
    768. 

  768. <a class="sourceLine" id="cb55-3" title="3"><span class="bu">print</span>(<span class="ss">f&#39;The dot product of v1 and v2 = </span><span class="sc">{np.</span>dot(v1, v2)<span class="sc">}</span><span class="ss">&#39;</span>)</a></code></pre></div>
    769. 

  769. <pre><code>The dot product of v1 and v2 = 4</code></pre>
    770. 

  770. <div class="sourceCode" id="cb57"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb57-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    771. 

  771. <a class="sourceLine" id="cb57-2" title="2"></a>
    772. 

  772. <a class="sourceLine" id="cb57-3" title="3"><span class="bu">print</span>(<span class="ss">f&#39;The dot product of v1 and v2 = </span><span class="sc">{np.</span>dot(np.array(v1), np.array(v2))<span class="sc">}</span><span class="ss">&#39;</span>)</a></code></pre></div>
    773. 

  773. <pre><code>The dot product of v1 and v2 = 4</code></pre>
    774. 

  774. <div class="sourceCode" id="cb59"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb59-1" title="1">va <span class="op">=</span> [<span class="dv">3</span>, <span class="dv">4</span>]</a>
    775. 

  775. <a class="sourceLine" id="cb59-2" title="2"></a>
    776. 

  776. <a class="sourceLine" id="cb59-3" title="3"><span class="bu">print</span>(<span class="ss">f&#39;The length of va = </span><span class="sc">{</span>(np.dot(va, va))<span class="op">**</span><span class="fl">0.5</span><span class="sc">}</span><span class="ss">&#39;</span>)</a></code></pre></div>
    777. 

  777. <pre><code>The length of va = 5.0</code></pre>
    778. 

  778. <h2 id="proof-of-the-cauchy-schwarz-inequality-with-code">Proof Of The Cauchy-Schwarz Inequality With Code</h2>
    779. 

  779. <p>Let’s first review the Cauchy-Schwarz Inequality with math …</p>
    780. 

  780. <p><br /><span class="math display">∥<em>x⃗</em>∥∥<em>y⃗</em>∥ ≥ |<em>x⃗</em> ⋅ <em>y⃗</em>|</span><br /></p>
    781. 

  781. <p>Let’s think about this together. The Cauchy-Schwarz Inequality is saying that the product of the magnitudes of two vectors is always greater than or equal to the absolute value of the dot product of those two vectors.</p>
    782. 

  782. <p>Let’s think about it even more. If the two vectors have the same direction, such as the angle between them is 0 degrees, then the product of their magnitudes will be the same as the absolute value of their dot products. If their is even a slight angle between the two vectors, then the product of their magnitudes will be greater than the absolute value of their dot products.</p>
    783. 

  783. <p>Sal does a great job of proving this in his lecture on the Cauchy-Schwarz Inequality. Let’s do a pseudo proof with code.</p>
    784. 

  784. <p>Let’s create a function that will check the Cauchy-Schwarz inequality for any given two vectors.</p>
    785. 

  785. <div class="sourceCode" id="cb61"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb61-1" title="1"><span class="kw">def</span> cauchy_schwarz_inequality(va, vb):</a>
    786. 

  786. <a class="sourceLine" id="cb61-2" title="2">    <span class="cf">assert</span> <span class="bu">len</span>(va) <span class="op">==</span> <span class="bu">len</span>(vb), <span class="st">&quot;Vectors are not the same dimension&quot;</span></a>
    787. 

  787. <a class="sourceLine" id="cb61-3" title="3">    va_norm <span class="op">=</span> np.linalg.norm(va)</a>
    788. 

  788. <a class="sourceLine" id="cb61-4" title="4">    vb_norm <span class="op">=</span> np.linalg.norm(vb)</a>
    789. 

  789. <a class="sourceLine" id="cb61-5" title="5">    abs_va_dot_vb <span class="op">=</span> np.<span class="bu">abs</span>(np.dot(va, vb))</a>
    790. 

  790. <a class="sourceLine" id="cb61-6" title="6"></a>
    791. 

  791. <a class="sourceLine" id="cb61-7" title="7">    the_check <span class="op">=</span> <span class="bu">round</span>(va_norm <span class="op">*</span> vb_norm, <span class="dv">12</span>) <span class="op">&gt;=</span> <span class="bu">round</span>(abs_va_dot_vb, <span class="dv">12</span>)</a>
    792. 

  792. <a class="sourceLine" id="cb61-8" title="8">    <span class="cf">if</span> <span class="kw">not</span> the_check:</a>
    793. 

  793. <a class="sourceLine" id="cb61-9" title="9">        <span class="bu">print</span>(va_norm <span class="op">*</span> vb_norm)</a>
    794. 

  794. <a class="sourceLine" id="cb61-10" title="10">        <span class="bu">print</span>(abs_va_dot_vb)</a>
    795. 

  795. <a class="sourceLine" id="cb61-11" title="11">        <span class="cf">return</span> <span class="st">&quot;Cauchy-Schwarz Inequality Failed!&quot;</span></a>
    796. 

  796. <a class="sourceLine" id="cb61-12" title="12">    <span class="cf">else</span>:</a>
    797. 

  797. <a class="sourceLine" id="cb61-13" title="13">        <span class="cf">return</span> <span class="st">&quot;Cauchy-Schwarz Inequality Holds!&quot;</span></a></code></pre></div>
    798. 

  798. <div class="sourceCode" id="cb62"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb62-1" title="1">count <span class="op">=</span> <span class="dv">0</span></a>
    799. 

  799. <a class="sourceLine" id="cb62-2" title="2">finite_span <span class="op">=</span> <span class="dv">10</span></a>
    800. 

  800. <a class="sourceLine" id="cb62-3" title="3">did_not_hold <span class="op">=</span> <span class="va">False</span></a>
    801. 

  801. <a class="sourceLine" id="cb62-4" title="4"></a>
    802. 

  802. <a class="sourceLine" id="cb62-5" title="5"><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>):</a>
    803. 

  803. <a class="sourceLine" id="cb62-6" title="6">    <span class="cf">for</span> j <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>):</a>
    804. 

  804. <a class="sourceLine" id="cb62-7" title="7">        <span class="cf">for</span> k <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>):</a>
    805. 

  805. <a class="sourceLine" id="cb62-8" title="8">            <span class="cf">for</span> l <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span>finite_span, finite_span<span class="op">+</span><span class="dv">1</span>):</a>
    806. 

  806. <a class="sourceLine" id="cb62-9" title="9">                count <span class="op">+=</span> <span class="dv">1</span></a>
    807. 

  807. <a class="sourceLine" id="cb62-10" title="10">                result <span class="op">=</span> cauchy_schwarz_inequality([i, j], [k, l])</a>
    808. 

  808. <a class="sourceLine" id="cb62-11" title="11">                <span class="cf">if</span> result <span class="op">!=</span> <span class="st">&quot;Cauchy-Schwarz Inequality Holds!&quot;</span>:</a>
    809. 

  809. <a class="sourceLine" id="cb62-12" title="12">                    did_not_hold <span class="op">=</span> <span class="va">True</span></a>
    810. 

  810. <a class="sourceLine" id="cb62-13" title="13"></a>
    811. 

  811. <a class="sourceLine" id="cb62-14" title="14"><span class="cf">if</span> did_not_hold:</a>
    812. 

  812. <a class="sourceLine" id="cb62-15" title="15">    <span class="bu">print</span>(<span class="st">&quot;Cauchy-Schwarz Inequality Failed!&quot;</span>)</a>
    813. 

  813. <a class="sourceLine" id="cb62-16" title="16"><span class="cf">else</span>:</a>
    814. 

  814. <a class="sourceLine" id="cb62-17" title="17">    <span class="bu">print</span>(<span class="ss">f&quot;Cauchy-Schwarz Inequality held for </span><span class="sc">{</span>count<span class="sc">}</span><span class="ss"> tests&quot;</span>)</a></code></pre></div>
    815. 

  815. <pre><code>Cauchy-Schwarz Inequality held for 194481 tests</code></pre>
    816. 

  816. <h2 id="vpython-code-to-run-from-an-ide">VPython Code To Run From An IDE</h2>
    817. 

  817. <div class="sourceCode" id="cb64"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb64-1" title="1"><span class="co"># You MIGHT be able to get this to run in Colab, but</span></a>
    818. 

  818. <a class="sourceLine" id="cb64-2" title="2"><span class="co"># I have not yet done this</span></a>
    819. 

  819. <a class="sourceLine" id="cb64-3" title="3"><span class="im">import</span> vpython <span class="im">as</span> vp</a>
    820. 

  820. <a class="sourceLine" id="cb64-4" title="4"></a>
    821. 

  821. <a class="sourceLine" id="cb64-5" title="5">floor<span class="op">=</span>vp.box(pos<span class="op">=</span>vp.vector(<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">0</span>),</a>
    822. 

  822. <a class="sourceLine" id="cb64-6" title="6">              size<span class="op">=</span>vp.vector(<span class="dv">12</span>,<span class="fl">0.5</span>,<span class="dv">12</span>),</a>
    823. 

  823. <a class="sourceLine" id="cb64-7" title="7">              color<span class="op">=</span>vp.color.red)</a>
    824. 

  824. <a class="sourceLine" id="cb64-8" title="8"></a>
    825. 

  825. <a class="sourceLine" id="cb64-9" title="9">ball<span class="op">=</span>vp.sphere(pos<span class="op">=</span>vp.vector(<span class="dv">0</span>,<span class="dv">7</span>,<span class="dv">0</span>),</a>
    826. 

  826. <a class="sourceLine" id="cb64-10" title="10">               radius<span class="op">=</span><span class="dv">2</span>,</a>
    827. 

  827. <a class="sourceLine" id="cb64-11" title="11">               color<span class="op">=</span>vp.color.blue)</a>
    828. 

  828. <a class="sourceLine" id="cb64-12" title="12"></a>
    829. 

  829. <a class="sourceLine" id="cb64-13" title="13">ball.velocity <span class="op">=</span> vp.vector(<span class="dv">0</span>,<span class="op">-</span><span class="dv">1</span>,<span class="dv">0</span>)</a>
    830. 

  830. <a class="sourceLine" id="cb64-14" title="14">dt <span class="op">=</span> <span class="fl">0.01</span></a>
    831. 

  831. <a class="sourceLine" id="cb64-15" title="15"></a>
    832. 

  832. <a class="sourceLine" id="cb64-16" title="16"><span class="cf">while</span> <span class="va">True</span>:</a>
    833. 

  833. <a class="sourceLine" id="cb64-17" title="17">    vp.rate(<span class="dv">100</span>)</a>
    834. 

  834. <a class="sourceLine" id="cb64-18" title="18">    ball.pos <span class="op">=</span> ball.pos <span class="op">+</span> ball.velocity <span class="op">*</span> dt</a>
    835. 

  835. <a class="sourceLine" id="cb64-19" title="19">    <span class="cf">if</span> ball.pos.y <span class="op">&lt;</span> (ball.radius<span class="op">+</span>floor.size.y<span class="op">/</span><span class="dv">2</span>):</a>
    836. 

  836. <a class="sourceLine" id="cb64-20" title="20">        ball.velocity.y <span class="op">=</span> <span class="bu">abs</span>(ball.velocity.y)</a>
    837. 

  837. <a class="sourceLine" id="cb64-21" title="21">    <span class="cf">else</span>:</a>
    838. 

  838. <a class="sourceLine" id="cb64-22" title="22">        ball.velocity.y <span class="op">=</span> ball.velocity.y <span class="op">-</span> <span class="fl">9.8</span> <span class="op">*</span> dt</a></code></pre></div>
    839. 

  839. <h2 id="vector-triangle-inequality">Vector Triangle Inequality</h2>
    840. 

  840. <p>Cauchy-Schwarz Inequality Repeated</p>
    841. 

  841. <p>For <br /><span class="math display"><em>x⃗</em>  <em>a</em><em>n</em><em>d</em>  <em>y⃗</em> ∈ <em>ℝ</em><sup><em>n</em></sup></span><br /></p>
    842. 

  842. <p>with $  ;; and ;;  $ as non-zero vectors,</p>
    843. 

  843. <p><br /><span class="math display">∥<em>x⃗</em>∥∥<em>y⃗</em>∥ ≥ |<em>x⃗</em> ⋅ <em>y⃗</em>|</span><br /></p>
    844. 

  844. <p>Note too that if c is non zero, then $;;  ; = ; c ;  $ $ $ $  =   $</p>
    845. 

  845. <p>Let’s experiment. Remember that the length of a vector squared is equal to the dot product of that vector with itself.</p>
    846. 

  846. <p>Let’s remind ourselves that</p>
    847. 

  847. <p><br /><span class="math display">∥<em>a⃗</em>∥<sup>2</sup> = <em>a⃗</em> ⋅ <em>a⃗</em></span><br /></p>
    848. 

  848. <p>So, let’s try something interesting …</p>
    849. 

  849. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup> = (<em>x⃗</em> + <em>y⃗</em>) ⋅ (<em>x⃗</em> + <em>y⃗</em>)</span><br /></p>
    850. 

  850. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup> = (<em>x⃗</em> ⋅ <em>x⃗</em>) + 2(<em>x⃗</em> ⋅ <em>y⃗</em>) + (<em>y⃗</em> ⋅ <em>y⃗</em>)</span><br /></p>
    851. 

  851. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup> = ∥<em>x⃗</em>∥<sup>2</sup> + 2(<em>x⃗</em> ⋅ <em>y⃗</em>) + ∥<em>y⃗</em>∥<sup>2</sup></span><br /></p>
    852. 

  852. <p>Now remember that</p>
    853. 

  853. <p><br /><span class="math display">|<em>x⃗</em> ⋅ <em>y⃗</em>| ≥ <em>x⃗</em> ⋅ <em>y⃗</em></span><br /></p>
    854. 

  854. <p>and thus</p>
    855. 

  855. <p><br /><span class="math display">∥<em>x⃗</em>∥∥<em>y⃗</em>∥ ≥ |<em>x⃗</em> ⋅ <em>y⃗</em>| ≥ <em>x⃗</em> ⋅ <em>y⃗</em></span><br /></p>
    856. 

  856. <p>Therefore,</p>
    857. 

  857. <p><br /><span class="math display">∥<em>x⃗</em>∥<sup>2</sup> + 2∥<em>x⃗</em>∥∥<em>y⃗</em>∥ + ∥<em>y⃗</em>∥<sup>2</sup> ≥ ∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup></span><br /></p>
    858. 

  858. <p>or</p>
    859. 

  859. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup> ≤ ∥<em>x⃗</em>∥<sup>2</sup> + 2∥<em>x⃗</em>∥∥<em>y⃗</em>∥ + ∥<em>y⃗</em>∥<sup>2</sup></span><br /></p>
    860. 

  860. <p>But, looking at the right, we see that it is a square. So,</p>
    861. 

  861. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥<sup>2</sup> ≤ (∥<em>x⃗</em>∥ + ∥<em>y⃗</em>∥)<sup>2</sup></span><br /></p>
    862. 

  862. <p>Now, let’s take the square root of both sides …</p>
    863. 

  863. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥ ≤ ∥<em>x⃗</em>∥ + ∥<em>y⃗</em>∥</span><br /></p>
    864. 

  864. <p>This is called <strong>The Triangle Inequality</strong></p>
    865. 

  865. <p>So now it’s time for our Python and Matplotlib graph paper!</p>
    866. 

  866. <div class="sourceCode" id="cb65"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb65-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    867. 

  867. <a class="sourceLine" id="cb65-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    868. 

  868. <a class="sourceLine" id="cb65-3" title="3"></a>
    869. 

  869. <a class="sourceLine" id="cb65-4" title="4">plt.rc_context({  <span class="co"># Only needed for some dark modes</span></a>
    870. 

  870. <a class="sourceLine" id="cb65-5" title="5">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    871. 

  871. <a class="sourceLine" id="cb65-6" title="6">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    872. 

  872. <a class="sourceLine" id="cb65-7" title="7"></a>
    873. 

  873. <a class="sourceLine" id="cb65-8" title="8"><span class="co"># Create a pallet plot for vectors</span></a>
    874. 

  874. <a class="sourceLine" id="cb65-9" title="9">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    875. 

  875. <a class="sourceLine" id="cb65-10" title="10">plt.xlim([<span class="op">-</span><span class="dv">3</span>, <span class="dv">3</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">1</span>, <span class="dv">7</span>])</a>
    876. 

  876. <a class="sourceLine" id="cb65-11" title="11"><span class="co"># plt.xlim([-5, 3]); plt.ylim([-1, 7])</span></a>
    877. 

  877. <a class="sourceLine" id="cb65-12" title="12">plt.grid()</a>
    878. 

  878. <a class="sourceLine" id="cb65-13" title="13"></a>
    879. 

  879. <a class="sourceLine" id="cb65-14" title="14">x <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">4</span>]<span class="op">;</span> y <span class="op">=</span> [<span class="op">-</span><span class="dv">4</span>, <span class="dv">2</span>]</a>
    880. 

  880. <a class="sourceLine" id="cb65-15" title="15"></a>
    881. 

  881. <a class="sourceLine" id="cb65-16" title="16">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    882. 

  882. <a class="sourceLine" id="cb65-17" title="17">          ec <span class="op">=</span><span class="st">&#39;green&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    883. 

  883. <a class="sourceLine" id="cb65-18" title="18">plt.arrow(<span class="dv">2</span>, <span class="dv">4</span>, y[<span class="dv">0</span>], y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    884. 

  884. <a class="sourceLine" id="cb65-19" title="19">          ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    885. 

  885. <a class="sourceLine" id="cb65-20" title="20"><span class="co"># plt.arrow(0, 0, y[0], y[1], head_width = 0.2, width = 0.05,</span></a>
    886. 

  886. <a class="sourceLine" id="cb65-21" title="21"><span class="co">#           ec =&#39;red&#39;, length_includes_head=True)</span></a>
    887. 

  887. <a class="sourceLine" id="cb65-22" title="22"></a>
    888. 

  888. <a class="sourceLine" id="cb65-23" title="23">x_plus_y <span class="op">=</span> np.array(x) <span class="op">+</span> np.array(y)</a>
    889. 

  889. <a class="sourceLine" id="cb65-24" title="24"></a>
    890. 

  890. <a class="sourceLine" id="cb65-25" title="25">plt.text(<span class="dv">1</span>, <span class="dv">1</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    891. 

  891. <a class="sourceLine" id="cb65-26" title="26">plt.text(<span class="dv">0</span>, <span class="fl">5.5</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    892. 

  892. <a class="sourceLine" id="cb65-27" title="27">plt.text(<span class="op">-</span><span class="dv">2</span>, <span class="fl">2.5</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs"> + \vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    893. 

  893. <a class="sourceLine" id="cb65-28" title="28"></a>
    894. 

  894. <a class="sourceLine" id="cb65-29" title="29">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x_plus_y[<span class="dv">0</span>], x_plus_y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>,</a>
    895. 

  895. <a class="sourceLine" id="cb65-30" title="30">          width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;blue&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    896. 

  896. <a class="sourceLine" id="cb65-31" title="31">plt.show()<span class="op">;</span></a></code></pre></div>
    897. 

  897. <figure>
    898. 

  898. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_89_0.png" alt="png" /><figcaption>png</figcaption>
    899. 

  899. </figure>
    900. 

  900. <p>Now, what is the extreme case where</p>
    901. 

  901. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥ = ∥<em>x⃗</em>∥ + ∥<em>y⃗</em>∥</span><br /></p>
    902. 

  902. <p>It must be when $  ;; and ;;  $ are pointing in the same direction. If they are in opposite directions, the minimum of an inequality happens on the right hand side.</p>
    903. 

  903. <div class="sourceCode" id="cb66"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb66-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    904. 

  904. <a class="sourceLine" id="cb66-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    905. 

  905. <a class="sourceLine" id="cb66-3" title="3"></a>
    906. 

  906. <a class="sourceLine" id="cb66-4" title="4"><span class="co"># Create a pallet plot for vectors</span></a>
    907. 

  907. <a class="sourceLine" id="cb66-5" title="5">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    908. 

  908. <a class="sourceLine" id="cb66-6" title="6">plt.xlim([<span class="op">-</span><span class="dv">7</span>, <span class="dv">1</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">1</span>, <span class="dv">7</span>])</a>
    909. 

  909. <a class="sourceLine" id="cb66-7" title="7"><span class="co"># plt.xlim([-3, 3]); plt.ylim([-3, 3])</span></a>
    910. 

  910. <a class="sourceLine" id="cb66-8" title="8">plt.grid()</a>
    911. 

  911. <a class="sourceLine" id="cb66-9" title="9"></a>
    912. 

  912. <a class="sourceLine" id="cb66-10" title="10">x <span class="op">=</span> [<span class="op">-</span><span class="dv">2</span>, <span class="dv">2</span>]<span class="op">;</span> y <span class="op">=</span> [<span class="op">-</span><span class="dv">4</span>, <span class="dv">4</span>]</a>
    913. 

  913. <a class="sourceLine" id="cb66-11" title="11"><span class="co"># x = [2, 2]; y = [-4, -4]</span></a>
    914. 

  914. <a class="sourceLine" id="cb66-12" title="12"></a>
    915. 

  915. <a class="sourceLine" id="cb66-13" title="13">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    916. 

  916. <a class="sourceLine" id="cb66-14" title="14">          ec <span class="op">=</span><span class="st">&#39;green&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    917. 

  917. <a class="sourceLine" id="cb66-15" title="15">plt.arrow(x[<span class="dv">0</span>]<span class="op">-</span><span class="fl">0.1</span>, x[<span class="dv">1</span>]<span class="op">+</span><span class="fl">0.1</span>, y[<span class="dv">0</span>], y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    918. 

  918. <a class="sourceLine" id="cb66-16" title="16">          ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    919. 

  919. <a class="sourceLine" id="cb66-17" title="17"></a>
    920. 

  920. <a class="sourceLine" id="cb66-18" title="18">x_plus_y <span class="op">=</span> np.array(x) <span class="op">+</span> np.array(y)</a>
    921. 

  921. <a class="sourceLine" id="cb66-19" title="19"></a>
    922. 

  922. <a class="sourceLine" id="cb66-20" title="20">plt.text(<span class="op">-</span><span class="fl">1.5</span>, <span class="fl">0.5</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    923. 

  923. <a class="sourceLine" id="cb66-21" title="21">plt.text(<span class="op">-</span><span class="dv">4</span>, <span class="fl">3.0</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    924. 

  924. <a class="sourceLine" id="cb66-22" title="22">plt.text(<span class="op">-</span><span class="dv">3</span>, <span class="fl">3.7</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs"> + \vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    925. 

  925. <a class="sourceLine" id="cb66-23" title="23"></a>
    926. 

  926. <a class="sourceLine" id="cb66-24" title="24"><span class="co"># </span><span class="al">NOTE</span><span class="co"> that we are offsetting by the 2 * width of arrows to see all vectors</span></a>
    927. 

  927. <a class="sourceLine" id="cb66-25" title="25">plt.arrow(<span class="fl">0.1</span>, <span class="fl">0.1</span>, x_plus_y[<span class="dv">0</span>], x_plus_y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>,</a>
    928. 

  928. <a class="sourceLine" id="cb66-26" title="26">          width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;blue&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    929. 

  929. <a class="sourceLine" id="cb66-27" title="27"><span class="co"># plt.arrow(0.1, -0.1, x_plus_y[0], x_plus_y[1], head_width = 0.2,</span></a>
    930. 

  930. <a class="sourceLine" id="cb66-28" title="28"><span class="co">#           width = 0.05, ec =&#39;blue&#39;, length_includes_head=True)</span></a>
    931. 

  931. <a class="sourceLine" id="cb66-29" title="29">plt.show()<span class="op">;</span></a></code></pre></div>
    932. 

  932. <figure>
    933. 

  933. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_91_0.png" alt="png" /><figcaption>png</figcaption>
    934. 

  934. </figure>
    935. 

  935. <p>We find a minimum for the right hand side of</p>
    936. 

  936. <p><br /><span class="math display">∥<em>x⃗</em> + <em>y⃗</em>∥ ≤ ∥<em>x⃗</em>∥ + ∥<em>y⃗</em>∥</span><br /></p>
    937. 

  937. <p>when the vectors point in <strong>perfectly</strong> opposite directions (they are $ 180^{} $ with respect to each other).</p>
    938. 

  938. <div class="sourceCode" id="cb67"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb67-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    939. 

  939. <a class="sourceLine" id="cb67-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    940. 

  940. <a class="sourceLine" id="cb67-3" title="3"></a>
    941. 

  941. <a class="sourceLine" id="cb67-4" title="4"><span class="co"># Create a pallet plot for vectors</span></a>
    942. 

  942. <a class="sourceLine" id="cb67-5" title="5">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    943. 

  943. <a class="sourceLine" id="cb67-6" title="6"><span class="co"># plt.xlim([-7, 1]); plt.ylim([-1, 7])</span></a>
    944. 

  944. <a class="sourceLine" id="cb67-7" title="7">plt.xlim([<span class="op">-</span><span class="dv">3</span>, <span class="dv">3</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">3</span>, <span class="dv">3</span>])</a>
    945. 

  945. <a class="sourceLine" id="cb67-8" title="8">plt.grid()</a>
    946. 

  946. <a class="sourceLine" id="cb67-9" title="9"></a>
    947. 

  947. <a class="sourceLine" id="cb67-10" title="10"><span class="co"># x = [-2, 2]; y = [-4, 4]</span></a>
    948. 

  948. <a class="sourceLine" id="cb67-11" title="11">x <span class="op">=</span> [<span class="dv">2</span>, <span class="dv">2</span>]<span class="op">;</span> y <span class="op">=</span> [<span class="op">-</span><span class="dv">4</span>, <span class="dv">-4</span>]</a>
    949. 

  949. <a class="sourceLine" id="cb67-12" title="12"></a>
    950. 

  950. <a class="sourceLine" id="cb67-13" title="13">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    951. 

  951. <a class="sourceLine" id="cb67-14" title="14">          ec <span class="op">=</span><span class="st">&#39;green&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    952. 

  952. <a class="sourceLine" id="cb67-15" title="15">plt.arrow(x[<span class="dv">0</span>]<span class="op">-</span><span class="fl">0.1</span>, x[<span class="dv">1</span>]<span class="op">+</span><span class="fl">0.1</span>, y[<span class="dv">0</span>], y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    953. 

  953. <a class="sourceLine" id="cb67-16" title="16">          ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    954. 

  954. <a class="sourceLine" id="cb67-17" title="17"></a>
    955. 

  955. <a class="sourceLine" id="cb67-18" title="18">x_plus_y <span class="op">=</span> np.array(x) <span class="op">+</span> np.array(y)</a>
    956. 

  956. <a class="sourceLine" id="cb67-19" title="19"></a>
    957. 

  957. <a class="sourceLine" id="cb67-20" title="20">plt.text(<span class="op">-</span><span class="fl">1.5</span>, <span class="fl">0.5</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    958. 

  958. <a class="sourceLine" id="cb67-21" title="21">plt.text(<span class="op">-</span><span class="dv">4</span>, <span class="fl">3.0</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    959. 

  959. <a class="sourceLine" id="cb67-22" title="22">plt.text(<span class="op">-</span><span class="dv">3</span>, <span class="fl">3.7</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs"> + \vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    960. 

  960. <a class="sourceLine" id="cb67-23" title="23"></a>
    961. 

  961. <a class="sourceLine" id="cb67-24" title="24"><span class="co"># </span><span class="al">NOTE</span><span class="co"> that we are offsetting by the 2 * width of arrows to see all vectors</span></a>
    962. 

  962. <a class="sourceLine" id="cb67-25" title="25"><span class="co"># plt.arrow(0.1, 0.1, x_plus_y[0], x_plus_y[1], head_width = 0.2,</span></a>
    963. 

  963. <a class="sourceLine" id="cb67-26" title="26"><span class="co">#           width = 0.05, ec =&#39;blue&#39;, length_includes_head=True)</span></a>
    964. 

  964. <a class="sourceLine" id="cb67-27" title="27">plt.arrow(<span class="fl">0.1</span>, <span class="fl">-0.1</span>, x_plus_y[<span class="dv">0</span>], x_plus_y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>,</a>
    965. 

  965. <a class="sourceLine" id="cb67-28" title="28">          width <span class="op">=</span> <span class="fl">0.05</span>, ec <span class="op">=</span><span class="st">&#39;blue&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    966. 

  966. <a class="sourceLine" id="cb67-29" title="29">plt.show()<span class="op">;</span></a></code></pre></div>
    967. 

  967. <figure>
    968. 

  968. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_93_0.png" alt="png" /><figcaption>png</figcaption>
    969. 

  969. </figure>
    970. 

  970. <p>Finally, what is ULTRA cool is that this works for $ ℝ^n $!</p>
    971. 

  971. <h2 id="our-dynamic-experiment">Our Dynamic Experiment</h2>
    972. 

  972. <div class="sourceCode" id="cb68"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb68-1" title="1"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    973. 

  973. <a class="sourceLine" id="cb68-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    974. 

  974. <a class="sourceLine" id="cb68-3" title="3"><span class="im">from</span> math <span class="im">import</span> cos, sin, acos, pi</a>
    975. 

  975. <a class="sourceLine" id="cb68-4" title="4"></a>
    976. 

  976. <a class="sourceLine" id="cb68-5" title="5">plt.rc_context({  <span class="co"># Only needed for some dark modes</span></a>
    977. 

  977. <a class="sourceLine" id="cb68-6" title="6">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    978. 

  978. <a class="sourceLine" id="cb68-7" title="7">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a>
    979. 

  979. <a class="sourceLine" id="cb68-8" title="8"></a>
    980. 

  980. <a class="sourceLine" id="cb68-9" title="9"><span class="co"># Create a pallet plot for vectors</span></a>
    981. 

  981. <a class="sourceLine" id="cb68-10" title="10">plt.figure(figsize<span class="op">=</span>(<span class="dv">10</span>, <span class="dv">10</span>))</a>
    982. 

  982. <a class="sourceLine" id="cb68-11" title="11">plt.plot([<span class="dv">0</span>, <span class="fl">0.001</span>],[<span class="dv">0</span>, <span class="fl">0.001</span>], c<span class="op">=</span><span class="st">&#39;white&#39;</span>)</a>
    983. 

  983. <a class="sourceLine" id="cb68-12" title="12">plt.xlim([<span class="op">-</span><span class="dv">3</span>, <span class="dv">7</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">1</span>, <span class="dv">10</span>])</a>
    984. 

  984. <a class="sourceLine" id="cb68-13" title="13">plt.grid()</a>
    985. 

  985. <a class="sourceLine" id="cb68-14" title="14"></a>
    986. 

  986. <a class="sourceLine" id="cb68-15" title="15">xmag <span class="op">=</span> <span class="dv">5</span><span class="op">;</span> ymag <span class="op">=</span> <span class="dv">5</span><span class="op">;</span> x <span class="op">=</span> [<span class="dv">3</span>, <span class="dv">4</span>]</a>
    987. 

  987. <a class="sourceLine" id="cb68-16" title="16"></a>
    988. 

  988. <a class="sourceLine" id="cb68-17" title="17">align_dir <span class="op">=</span> acos(<span class="dv">3</span><span class="op">/</span><span class="dv">5</span>)<span class="op">/</span>pi<span class="op">*</span><span class="dv">180</span></a>
    989. 

  989. <a class="sourceLine" id="cb68-18" title="18">Angles <span class="op">=</span> np.arange(<span class="dv">0</span>, <span class="dv">180</span> <span class="op">+</span> <span class="dv">1</span>, <span class="dv">30</span>)</a>
    990. 

  990. <a class="sourceLine" id="cb68-19" title="19"></a>
    991. 

  991. <a class="sourceLine" id="cb68-20" title="20">plt.arrow(<span class="dv">0</span>, <span class="dv">0</span>, x[<span class="dv">0</span>], x[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    992. 

  992. <a class="sourceLine" id="cb68-21" title="21">          ec <span class="op">=</span><span class="st">&#39;green&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    993. 

  993. <a class="sourceLine" id="cb68-22" title="22">plt.text(<span class="dv">2</span>, <span class="fl">1.8</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    994. 

  994. <a class="sourceLine" id="cb68-23" title="23"></a>
    995. 

  995. <a class="sourceLine" id="cb68-24" title="24"><span class="cf">for</span> angle <span class="kw">in</span> Angles:</a>
    996. 

  996. <a class="sourceLine" id="cb68-25" title="25">    rad <span class="op">=</span> (angle <span class="op">+</span> align_dir) <span class="op">/</span> <span class="dv">180</span> <span class="op">*</span> pi</a>
    997. 

  997. <a class="sourceLine" id="cb68-26" title="26">    trad <span class="op">=</span> (angle <span class="op">+</span> align_dir <span class="op">-</span> <span class="dv">6</span>) <span class="op">/</span> <span class="dv">180</span> <span class="op">*</span> pi</a>
    998. 

  998. <a class="sourceLine" id="cb68-27" title="27">    y <span class="op">=</span> [ymag<span class="op">*</span>cos(rad), ymag<span class="op">*</span>sin(rad)]</a>
    999. 

  999. <a class="sourceLine" id="cb68-28" title="28">    ty <span class="op">=</span> [ymag<span class="op">*</span>cos(trad), ymag<span class="op">*</span>sin(trad)]</a>
    1000. 

  1000. <a class="sourceLine" id="cb68-29" title="29"></a>
    1001. 

  1001. <a class="sourceLine" id="cb68-30" title="30">    plt.arrow(x[<span class="dv">0</span>], x[<span class="dv">1</span>], y[<span class="dv">0</span>], y[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    1002. 

  1002. <a class="sourceLine" id="cb68-31" title="31">              ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    1003. 

  1003. <a class="sourceLine" id="cb68-32" title="32"></a>
    1004. 

  1004. <a class="sourceLine" id="cb68-33" title="33">    z <span class="op">=</span> np.array(x) <span class="op">+</span> np.array(y)</a>
    1005. 

  1005. <a class="sourceLine" id="cb68-34" title="34">    tp <span class="op">=</span> np.array(x) <span class="op">+</span> np.array(ty)<span class="op">*</span><span class="fl">0.7</span></a>
    1006. 

  1006. <a class="sourceLine" id="cb68-35" title="35"></a>
    1007. 

  1007. <a class="sourceLine" id="cb68-36" title="36">    plt.text(tp[<span class="dv">0</span>], tp[<span class="dv">1</span>], <span class="vs">r&#39;$\vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;left&#39;</span>)</a>
    1008. 

  1008. <a class="sourceLine" id="cb68-37" title="37">    plt.text(z[<span class="dv">0</span>]<span class="op">*</span><span class="fl">0.9</span> <span class="op">-</span> <span class="fl">0.25</span>, z[<span class="dv">1</span>]<span class="op">*</span><span class="fl">0.9</span>, <span class="vs">r&#39;$\vec</span><span class="sc">{x}</span><span class="vs"> + \vec</span><span class="sc">{y}</span><span class="vs">$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>,</a>
    1009. 

  1009. <a class="sourceLine" id="cb68-38" title="38">             ha<span class="op">=</span><span class="st">&#39;right&#39;</span>)</a>
    1010. 

  1010. <a class="sourceLine" id="cb68-39" title="39">    plt.text(z[<span class="dv">0</span>]<span class="op">*</span><span class="fl">0.8</span> <span class="op">-</span> <span class="fl">0.2</span>, z[<span class="dv">1</span>]<span class="op">*</span><span class="fl">0.8</span> <span class="op">-</span> <span class="fl">0.25</span>,</a>
    1011. 

  1011. <a class="sourceLine" id="cb68-40" title="40">             <span class="ss">f&#39;</span><span class="sc">{</span><span class="bu">round</span>(angle, <span class="dv">2</span>)<span class="sc">}</span><span class="ss">&#39;</span><span class="op">+</span><span class="vs">r&#39;$^{\circ}$&#39;</span>, fontsize<span class="op">=</span><span class="dv">14</span>, ha<span class="op">=</span><span class="st">&#39;right&#39;</span>)</a>
    1012. 

  1012. <a class="sourceLine" id="cb68-41" title="41"></a>
    1013. 

  1013. <a class="sourceLine" id="cb68-42" title="42">    plt.arrow(<span class="fl">0.07</span>, <span class="fl">-0.07</span>, z[<span class="dv">0</span>], z[<span class="dv">1</span>], head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    1014. 

  1014. <a class="sourceLine" id="cb68-43" title="43">              ec <span class="op">=</span><span class="st">&#39;blue&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    1015. 

  1015. <a class="sourceLine" id="cb68-44" title="44">    <span class="bu">print</span>(<span class="ss">f&#39;x + y with y </span><span class="sc">{</span>angle<span class="sc">}</span><span class="ss"> degrees ccw from x = </span><span class="sc">{</span><span class="bu">round</span>(np.linalg.norm(z), <span class="dv">5</span>)<span class="sc">:.5f}</span><span class="ss">&#39;</span>)</a>
    1016. 

  1016. <a class="sourceLine" id="cb68-45" title="45"></a>
    1017. 

  1017. <a class="sourceLine" id="cb68-46" title="46"><span class="bu">print</span>()</a>
    1018. 

  1018. <a class="sourceLine" id="cb68-47" title="47">plt.show()<span class="op">;</span></a></code></pre></div>
    1019. 

  1019. <pre><code>x + y with y 0 degrees ccw from x = 10.00000
    1020. 

  1020. x + y with y 30 degrees ccw from x = 9.65926
    1021. 

  1021. x + y with y 60 degrees ccw from x = 8.66025
    1022. 

  1022. x + y with y 90 degrees ccw from x = 7.07107
    1023. 

  1023. x + y with y 120 degrees ccw from x = 5.00000
    1024. 

  1024. x + y with y 150 degrees ccw from x = 2.58819
    1025. 

  1025. x + y with y 180 degrees ccw from x = 0.00000</code></pre>
    1026. 

  1026. <figure>
    1027. 

  1027. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_96_1.png" alt="png" /><figcaption>png</figcaption>
    1028. 

  1028. </figure>
    1029. 

  1029. <h2 id="a-line-in-2d-ℝ2-using-linear-algebra">A Line In 2D, <span class="math inline"><em>ℝ</em><sup>2</sup></span>, Using Linear Algebra</h2>
    1030. 

  1030. <p>An equation that defines a line subspace in 2D space or <span class="math inline"><em>ℝ</em><sup>2</sup></span> is <br><br> <br /><span class="math display">$$ \begin{bmatrix} 
    1031. 

  1031. a_1 \\ a_2 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix}
    1032. 

  1032. = a_1 x + a_2 y = d $$</span><br /> <br> But what do we choose for $ a_1 $ and $ a_2 $? The vector normal to the line that we want.</p>
    1033. 

  1033. <p>If I want a slope of 1, the normal to that line is the vector</p>
    1034. 

  1034. <p><br /><span class="math display">$$ \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} = 
    1035. 

  1035. \begin{bmatrix} -1 \\ 1 \end{bmatrix} $$</span><br /></p>
    1036. 

  1036. <p>What is the value for <span class="math inline"><em>d</em></span>? A <span class="math inline"><em>y</em></span> intercept if we solve for <span class="math inline"><em>y</em></span> interms of <span class="math inline"><em>x</em></span>, and an <span class="math inline"><em>x</em></span> intercept if we solve for <span class="math inline"><em>x</em></span> in terms of <span class="math inline"><em>y</em></span>!</p>
    1037. 

  1037. <p><br /><span class="math display"><em>y</em> =  − <em>a</em><sub>1</sub>/<em>a</em><sub>2</sub><em>x</em> + <em>d</em>/<em>a</em><sub>2</sub> = <em>m</em><em>x</em> + <em>y</em><sub><em>i</em><em>n</em><em>t</em></sub></span><br /></p>
    1038. 

  1038. <p>Let’s see 3 examples of normal vectors and y intercepts:</p>
    1039. 

  1039. <table>
    1040. 

  1040. <thead>
    1041. 

  1041. <tr class="header">
    1042. 

  1042. <th>Normal</th>
    1043. 

  1043. <th>Y Intercept</th>
    1044. 

  1044. </tr>
    1045. 

  1045. </thead>
    1046. 

  1046. <tbody>
    1047. 

  1047. <tr class="odd">
    1048. 

  1048. <td>[-2, 1]</td>
    1049. 

  1049. <td>1</td>
    1050. 

  1050. </tr>
    1051. 

  1051. <tr class="even">
    1052. 

  1052. <td>[-1, 1]</td>
    1053. 

  1053. <td>0</td>
    1054. 

  1054. </tr>
    1055. 

  1055. <tr class="odd">
    1056. 

  1056. <td>[-1, 2]</td>
    1057. 

  1057. <td>-1</td>
    1058. 

  1058. </tr>
    1059. 

  1059. </tbody>
    1060. 

  1060. </table>
    1061. 

  1061. <div class="sourceCode" id="cb70"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb70-1" title="1">x <span class="op">=</span> []<span class="op">;</span> yofx_10 <span class="op">=</span> []<span class="op">;</span> yofx_21 <span class="op">=</span> []<span class="op">;</span> yofx_0p5m1 <span class="op">=</span> []</a>
    1062. 

  1062. <a class="sourceLine" id="cb70-2" title="2"></a>
    1063. 

  1063. <a class="sourceLine" id="cb70-3" title="3"><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span><span class="dv">60</span>, <span class="dv">61</span>, <span class="dv">2</span>):</a>
    1064. 

  1064. <a class="sourceLine" id="cb70-4" title="4">    x.append(i<span class="op">/</span><span class="dv">10</span>)</a>
    1065. 

  1065. <a class="sourceLine" id="cb70-5" title="5">    yofx_10.append(<span class="dv">1</span><span class="op">*</span>i<span class="op">/</span><span class="dv">10</span><span class="op">+</span><span class="dv">0</span>)</a>
    1066. 

  1066. <a class="sourceLine" id="cb70-6" title="6">    yofx_21.append(<span class="dv">2</span><span class="op">*</span>(i<span class="op">/</span><span class="dv">10</span>)<span class="op">+</span><span class="dv">1</span>)</a>
    1067. 

  1067. <a class="sourceLine" id="cb70-7" title="7">    yofx_0p5m1.append(<span class="fl">0.5</span><span class="op">*</span>(i<span class="op">/</span><span class="dv">10</span>)<span class="op">-</span><span class="dv">1</span>)</a></code></pre></div>
    1068. 

  1068. <div class="sourceCode" id="cb71"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb71-1" title="1"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    1069. 

  1069. <a class="sourceLine" id="cb71-2" title="2"></a>
    1070. 

  1070. <a class="sourceLine" id="cb71-3" title="3">plt.figure(figsize<span class="op">=</span>(<span class="dv">8</span>,<span class="dv">8</span>))</a>
    1071. 

  1071. <a class="sourceLine" id="cb71-4" title="4">plt.xlim([<span class="op">-</span><span class="dv">6</span>, <span class="dv">6</span>])<span class="op">;</span> plt.ylim([<span class="op">-</span><span class="dv">6</span>, <span class="dv">6</span>])</a>
    1072. 

  1072. <a class="sourceLine" id="cb71-5" title="5">plt.plot(x, yofx_10)<span class="op">;</span> plt.plot(x, yofx_21)<span class="op">;</span> plt.plot(x, yofx_0p5m1)</a>
    1073. 

  1073. <a class="sourceLine" id="cb71-6" title="6">plt.arrow(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">-1</span>, <span class="fl">0.5</span>, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    1074. 

  1074. <a class="sourceLine" id="cb71-7" title="7">              ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    1075. 

  1075. <a class="sourceLine" id="cb71-8" title="8">plt.arrow(<span class="dv">4</span>, <span class="dv">4</span>, <span class="dv">-1</span>, <span class="dv">1</span>, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    1076. 

  1076. <a class="sourceLine" id="cb71-9" title="9">              ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    1077. 

  1077. <a class="sourceLine" id="cb71-10" title="10">plt.arrow(<span class="dv">4</span>, <span class="dv">1</span>, <span class="fl">-0.5</span>, <span class="dv">1</span>, head_width <span class="op">=</span> <span class="fl">0.2</span>, width <span class="op">=</span> <span class="fl">0.05</span>,</a>
    1078. 

  1078. <a class="sourceLine" id="cb71-11" title="11">              ec <span class="op">=</span><span class="st">&#39;red&#39;</span>, length_includes_head<span class="op">=</span><span class="va">True</span>)</a>
    1079. 

  1079. <a class="sourceLine" id="cb71-12" title="12">plt.grid()</a>
    1080. 

  1080. <a class="sourceLine" id="cb71-13" title="13">plt.show()</a></code></pre></div>
    1081. 

  1081. <figure>
    1082. 

  1082. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_100_0.png" alt="png" /><figcaption>png</figcaption>
    1083. 

  1083. </figure>
    1084. 

  1084. <p>Can we also treat <span class="math inline">$\bf A$</span> as a vector?</p>
    1085. 

  1085. <div class="sourceCode" id="cb72"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb72-1" title="1">line <span class="op">=</span> [[<span class="op">-</span><span class="dv">3</span>, <span class="dv">-2</span>], [<span class="dv">0</span>, <span class="dv">1</span>], [<span class="dv">3</span>, <span class="dv">4</span>]]</a>
    1086. 

  1086. <a class="sourceLine" id="cb72-2" title="2">dfl <span class="op">=</span> pd.DataFrame(data<span class="op">=</span>line, columns<span class="op">=</span>[<span class="st">&#39;X&#39;</span>, <span class="st">&#39;Y&#39;</span>])</a>
    1087. 

  1087. <a class="sourceLine" id="cb72-3" title="3">dfl</a></code></pre></div>
    1088. 

  1088. <div id="df-ba983a39-0d36-49db-9c90-c6fed8fb1993">
    1089. 

  1089. <pre><code>&lt;div class=&quot;colab-df-container&quot;&gt;
    1090. 

  1090.   &lt;div&gt;</code></pre>
    1091. 

  1091. <style scoped>
    1092. 

  1092.     .dataframe tbody tr th:only-of-type {
    1093. 

  1093.         vertical-align: middle;
    1094. 

  1094.     }
    1095. 

  1095. 

  1096.     .dataframe tbody tr th {
    1097. 

  1097.         vertical-align: top;
    1098. 

  1098.     }
    1099. 

  1099. 

  1100.     .dataframe thead th {
    1101. 

  1101.         text-align: right;
    1102. 

  1102.     }
    1103. 

  1103. </style>
    1104. 

  1104. <table border="1" class="dataframe">
    1105. 

  1105. <thead>
    1106. 

  1106. <tr style="text-align: right;">
    1107. 

  1107. <th>
    1108. 

  1108. </th>
    1109. 

  1109. <th>
    1110. 

  1110. X
    1111. 

  1111. </th>
    1112. 

  1112. <th>
    1113. 

  1113. Y
    1114. 

  1114. </th>
    1115. 

  1115. </tr>
    1116. 

  1116. </thead>
    1117. 

  1117. <tbody>
    1118. 

  1118. <tr>
    1119. 

  1119. <th>
    1120. 

  1120. 0
    1121. 

  1121. </th>
    1122. 

  1122. <td>
    1123. 

  1123. -3
    1124. 

  1124. </td>
    1125. 

  1125. <td>
    1126. 

  1126. -2
    1127. 

  1127. </td>
    1128. 

  1128. </tr>
    1129. 

  1129. <tr>
    1130. 

  1130. <th>
    1131. 

  1131. 1
    1132. 

  1132. </th>
    1133. 

  1133. <td>
    1134. 

  1134. 0
    1135. 

  1135. </td>
    1136. 

  1136. <td>
    1137. 

  1137. 1
    1138. 

  1138. </td>
    1139. 

  1139. </tr>
    1140. 

  1140. <tr>
    1141. 

  1141. <th>
    1142. 

  1142. 2
    1143. 

  1143. </th>
    1144. 

  1144. <td>
    1145. 

  1145. 3
    1146. 

  1146. </td>
    1147. 

  1147. <td>
    1148. 

  1148. 4
    1149. 

  1149. </td>
    1150. 

  1150. </tr>
    1151. 

  1151. </tbody>
    1152. 

  1152. </table>
    1153. 

  1153. </div>
    1154. 

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    1155. 

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    1157. 

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    1158. 

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    1159. 

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    1160. 

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    1161. 

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    1162. 

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    1164. 

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    1179. 

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    1182. 

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    1186. 

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    1204. 

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    1210. 

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  1216.           element.appendChild(docLink);
    1217. 

  1217.         }
    1218. 

  1218.       </script>
    1219. 

  1219. </div>
    1220. 

  1220. </div>
    1221. 

  1221. <h2 id="the-equation-for-a-plane-in-3d-space---ℝ3">The Equation for a Plane in 3D Space - <span class="math inline"><em>ℝ</em><sup>3</sup></span></h2>
    1222. 

  1222. <p><br /><span class="math display">$$ \vec A \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} =
    1223. 

  1223. \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \cdot 
    1224. 

  1224. \begin{bmatrix} x \\ y \\ z \end{bmatrix} 
    1225. 

  1225. = a_1 x + a_2 y + a_3 z = d $$</span><br /> <br> where <span class="math inline">$\bf A$</span> is normal to the plane!</p>
    1226. 

  1226. <p>Now, IF you do NOT know <span class="math inline"><em>A</em></span>, but you know 3 points in the plane, you can solve for <span class="math inline"><em>A</em></span> this way …</p>
    1227. 

  1227. <p><br /><span class="math display">$$\begin{bmatrix} 
    1228. 

  1228. x_1 &amp; y_1 &amp; z_1 \\ x_2 &amp; y_2 &amp; z_2 \\ x_3 &amp; y_3 &amp; z_3
    1229. 

  1229. \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} = 
    1230. 

  1230. \begin{bmatrix} d \\ d \\ d \end{bmatrix}$$</span><br /> <br> We need at least 3 points in $ ℝ^3 $ to develop this equation for a plane. Once we have solved for $ \bf A$ as a vector?</p>
    1231. 

  1231. <div class="sourceCode" id="cb74"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb74-1" title="1">line <span class="op">=</span> [[<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>], [<span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>]]</a>
    1232. 

  1232. <a class="sourceLine" id="cb74-2" title="2">dfl <span class="op">=</span> pd.DataFrame(data<span class="op">=</span>line, columns<span class="op">=</span>[<span class="st">&#39;X&#39;</span>, <span class="st">&#39;Y&#39;</span>, <span class="st">&#39;Z&#39;</span>])</a>
    1233. 

  1233. <a class="sourceLine" id="cb74-3" title="3">dfl</a></code></pre></div>
    1234. 

  1234. <div id="df-86e22d6b-a3d1-419c-9f8d-219652fb460c">
    1235. 

  1235. <pre><code>&lt;div class=&quot;colab-df-container&quot;&gt;
    1236. 

  1236.   &lt;div&gt;</code></pre>
    1237. 

  1237. <style scoped>
    1238. 

  1238.     .dataframe tbody tr th:only-of-type {
    1239. 

  1239.         vertical-align: middle;
    1240. 

  1240.     }
    1241. 

  1241. 

  1242.     .dataframe tbody tr th {
    1243. 

  1243.         vertical-align: top;
    1244. 

  1244.     }
    1245. 

  1245. 

  1246.     .dataframe thead th {
    1247. 

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    1248. 

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    1249. 

  1249. </style>
    1250. 

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    1251. 

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    1252. 

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    1253. 

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    1254. 

  1254. </th>
    1255. 

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    1256. 

  1256. X
    1257. 

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    1258. 

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    1259. 

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    1260. 

  1260. </th>
    1261. 

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    1262. 

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    1264. 

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    1265. 

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    1266. 

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    1269. 

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    1283. 

  1283. 1
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    1286. 

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    1294. 

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    1295. 

  1295. </tbody>
    1296. 

  1296. </table>
    1297. 

  1297. </div>
    1298. 

  1298. <button class="colab-df-convert" onclick="convertToInteractive(&#39;df-86e22d6b-a3d1-419c-9f8d-219652fb460c&#39;)" title="Convert this dataframe to an interactive table." style="display:none;">
    1299. 

  1299. <svg xmlns="http://www.w3.org/2000/svg" height="24px"viewBox="0 0 24 24"
    1300. 

  1300.        width="24px"> <path d="M0 0h24v24H0V0z" fill="none"/> <path d="M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z"/><path d="M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z"/> </svg>
    1301. 

  1301. </button>
    1302. 

  1302. <style>
    1303. 

  1303.     .colab-df-container {
    1304. 

  1304.       display:flex;
    1305. 

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    1306. 

  1306.       gap: 12px;
    1307. 

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    1308. 

  1308. 

  1309.     .colab-df-convert {
    1310. 

  1310.       background-color: #E8F0FE;
    1311. 

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    1316. 

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    1317. 

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    1318. 

  1318.       width: 32px;
    1319. 

  1319.     }
    1320. 

  1320. 

  1321.     .colab-df-convert:hover {
    1322. 

  1322.       background-color: #E2EBFA;
    1323. 

  1323.       box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);
    1324. 

  1324.       fill: #174EA6;
    1325. 

  1325.     }
    1326. 

  1326. 

  1327.     [theme=dark] .colab-df-convert {
    1328. 

  1328.       background-color: #3B4455;
    1329. 

  1329.       fill: #D2E3FC;
    1330. 

  1330.     }
    1331. 

  1331. 

  1332.     [theme=dark] .colab-df-convert:hover {
    1333. 

  1333.       background-color: #434B5C;
    1334. 

  1334.       box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);
    1335. 

  1335.       filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));
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  1336.       fill: #FFFFFF;
    1337. 

  1337.     }
    1338. 

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    1339. 

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    1341. 

  1341.           document.querySelector('#df-86e22d6b-a3d1-419c-9f8d-219652fb460c button.colab-df-convert');
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  1342.         buttonEl.style.display =
    1343. 

  1343.           google.colab.kernel.accessAllowed ? 'block' : 'none';
    1344. 

  1344. 

  1345.         async function convertToInteractive(key) {
    1346. 

  1346.           const element = document.querySelector('#df-86e22d6b-a3d1-419c-9f8d-219652fb460c');
    1347. 

  1347.           const dataTable =
    1348. 

  1348.             await google.colab.kernel.invokeFunction('convertToInteractive',
    1349. 

  1349.                                                      [key], {});
    1350. 

  1350.           if (!dataTable) return;
    1351. 

  1351. 

  1352.           const docLinkHtml = 'Like what you see? Visit the ' +
    1353. 

  1353.             '<a target="_blank" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'
    1354. 

  1354.             + ' to learn more about interactive tables.';
    1355. 

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  1356.           dataTable['output_type'] = 'display_data';
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  1357.           await google.colab.output.renderOutput(dataTable, element);
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    1363. 

  1363. </div>
    1364. 

  1364. </div>
    1365. 

  1365. <div class="sourceCode" id="cb76"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb76-1" title="1"><span class="im">import</span> plotly.express <span class="im">as</span> px</a>
    1366. 

  1366. <a class="sourceLine" id="cb76-2" title="2"><span class="im">import</span> plotly.graph_objects <span class="im">as</span> go</a>
    1367. 

  1367. <a class="sourceLine" id="cb76-3" title="3"></a>
    1368. 

  1368. <a class="sourceLine" id="cb76-4" title="4">fig1 <span class="op">=</span> px.scatter_3d(dfp, x<span class="op">=</span><span class="st">&#39;X&#39;</span>, y<span class="op">=</span><span class="st">&#39;Y&#39;</span>, z<span class="op">=</span><span class="st">&#39;Z&#39;</span>, title<span class="op">=</span><span class="st">&quot;Title&quot;</span>, size<span class="op">=</span><span class="st">&#39;size&#39;</span>)</a>
    1369. 

  1369. <a class="sourceLine" id="cb76-5" title="5">fig2 <span class="op">=</span> px.line_3d(dfl, x<span class="op">=</span><span class="st">&#39;X&#39;</span>, y<span class="op">=</span><span class="st">&#39;Y&#39;</span>, z<span class="op">=</span><span class="st">&#39;Z&#39;</span>)</a>
    1370. 

  1370. <a class="sourceLine" id="cb76-6" title="6">fig3 <span class="op">=</span> go.Figure(data <span class="op">=</span> fig1.data <span class="op">+</span> fig2.data)</a>
    1371. 

  1371. <a class="sourceLine" id="cb76-7" title="7">fig3.show()</a></code></pre></div>
    1372. 

  1372. <html>
    1373. 

  1373. <head>
    1374. 

  1374. <meta charset="utf-8" />
    1375. 

  1375. </head>
    1376. 

  1376. <body>
    1377. 

  1377. <div>
    1378. 

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    1379. 

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                       {&quot;responsive&quot;: true}                    ).then(function(){</code></pre>
    1380. 

  1380. <p>var gd = document.getElementById(‘58dc9c1c-e06b-40fd-8759-1617a9c07e7a’); var x = new MutationObserver(function (mutations, observer) {{ var display = window.getComputedStyle(gd).display; if (!display || display === ‘none’) {{ console.log([gd, ‘removed!’]); Plotly.purge(gd); observer.disconnect(); }} }});</p>
    1381. 

  1381. <p>// Listen for the removal of the full notebook cells var notebookContainer = gd.closest(‘#notebook-container’); if (notebookContainer) {{ x.observe(notebookContainer, {childList: true}); }}</p>
    1382. 

  1382. <p>// Listen for the clearing of the current output cell var outputEl = gd.closest(‘.output’); if (outputEl) {{ x.observe(outputEl, {childList: true}); }}</p>
    1383. 

  1383. <pre><code>                    })                };                            &lt;/script&gt;        &lt;/div&gt;</code></pre>
    1384. 

  1384. </body>
    1385. 

  1385. </html>
    1386. 

  1386. <p>YES! In fact, <span class="math inline">$\bf A$</span> is the normal vector. Let’s test another case.</p>
    1387. 

  1387. <p>We can do the same with Matplotlib, but we won’t be able to rotate the plot in our notebook. Just run the same code on your PC.</p>
    1388. 

  1388. <div class="sourceCode" id="cb79"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb79-1" title="1"><span class="op">%</span>matplotlib inline</a></code></pre></div>
    1389. 

  1389. <div class="sourceCode" id="cb80"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb80-1" title="1"><span class="im">from</span> IPython.core.interactiveshell <span class="im">import</span> InteractiveShell</a>
    1390. 

  1390. <a class="sourceLine" id="cb80-2" title="2">InteractiveShell.ast_node_interactivity <span class="op">=</span> <span class="st">&quot;all&quot;</span></a></code></pre></div>
    1391. 

  1391. <div class="sourceCode" id="cb81"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb81-1" title="1">plt.rc_context({  <span class="co"># Only needed for some dark modes</span></a>
    1392. 

  1392. <a class="sourceLine" id="cb81-2" title="2">    <span class="st">&#39;axes.edgecolor&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;xtick.color&#39;</span>:<span class="st">&#39;black&#39;</span>,</a>
    1393. 

  1393. <a class="sourceLine" id="cb81-3" title="3">    <span class="st">&#39;ytick.color&#39;</span>:<span class="st">&#39;black&#39;</span>, <span class="st">&#39;figure.facecolor&#39;</span>:<span class="st">&#39;white&#39;</span>})<span class="op">;</span></a></code></pre></div>
    1394. 

  1394. <div class="sourceCode" id="cb82"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb82-1" title="1"><span class="im">from</span> mpl_toolkits.mplot3d <span class="im">import</span> axes3d</a>
    1395. 

  1395. <a class="sourceLine" id="cb82-2" title="2"><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</a>
    1396. 

  1396. <a class="sourceLine" id="cb82-3" title="3"><span class="im">from</span> matplotlib <span class="im">import</span> cm</a>
    1397. 

  1397. <a class="sourceLine" id="cb82-4" title="4"><span class="im">import</span> numpy <span class="im">as</span> np</a>
    1398. 

  1398. <a class="sourceLine" id="cb82-5" title="5"></a>
    1399. 

  1399. <a class="sourceLine" id="cb82-6" title="6">fig <span class="op">=</span> plt.figure()</a>
    1400. 

  1400. <a class="sourceLine" id="cb82-7" title="7">ax <span class="op">=</span> fig.add_subplot(<span class="dv">111</span>, projection<span class="op">=</span><span class="st">&#39;3d&#39;</span>)</a>
    1401. 

  1401. <a class="sourceLine" id="cb82-8" title="8"></a>
    1402. 

  1402. <a class="sourceLine" id="cb82-9" title="9">xy <span class="op">=</span> np.mgrid[<span class="op">-</span><span class="dv">2</span>:<span class="dv">2</span>:21j, <span class="dv">-2</span>:<span class="dv">2</span>:21j]</a>
    1403. 

  1403. <a class="sourceLine" id="cb82-10" title="10">z <span class="op">=</span> np.zeros((<span class="dv">21</span>, <span class="dv">21</span>))</a>
    1404. 

  1404. <a class="sourceLine" id="cb82-11" title="11">x <span class="op">=</span> xy[<span class="dv">0</span>]<span class="op">;</span> y <span class="op">=</span> xy[<span class="dv">1</span>]<span class="op">;</span> z <span class="op">=</span> <span class="dv">1</span> <span class="op">-</span> x <span class="op">-</span> y</a>
    1405. 

  1405. <a class="sourceLine" id="cb82-12" title="12">a <span class="op">=</span> [<span class="dv">0</span>, <span class="dv">1</span>]<span class="op">;</span> b <span class="op">=</span> [<span class="dv">0</span>, <span class="dv">1</span>]<span class="op">;</span> c <span class="op">=</span> [<span class="dv">0</span>, <span class="dv">1</span>]</a>
    1406. 

  1406. <a class="sourceLine" id="cb82-13" title="13"></a>
    1407. 

  1407. <a class="sourceLine" id="cb82-14" title="14">ax.plot_surface(x, y, z, cmap<span class="op">=</span>cm.coolwarm, linewidth<span class="op">=</span><span class="dv">0</span>, alpha<span class="op">=</span>.<span class="dv">7</span>)</a>
    1408. 

  1408. <a class="sourceLine" id="cb82-15" title="15">ax.plot(a, b, c, color<span class="op">=</span><span class="st">&quot;blue&quot;</span>)</a>
    1409. 

  1409. <a class="sourceLine" id="cb82-16" title="16"></a>
    1410. 

  1410. <a class="sourceLine" id="cb82-17" title="17">ax.set_title(<span class="st">&#39;A Plane in 3D Defined by a Normal Vector&#39;</span>)</a>
    1411. 

  1411. <a class="sourceLine" id="cb82-18" title="18">ax.set_xlabel(<span class="st">&#39;X&#39;</span>)<span class="op">;</span> ax.set_ylabel(<span class="st">&#39;Y&#39;</span>)<span class="op">;</span> ax.set_zlabel(<span class="st">&#39;Z&#39;</span>)</a>
    1412. 

  1412. <a class="sourceLine" id="cb82-19" title="19">ax.set_xlim(<span class="op">-</span><span class="dv">2</span>, <span class="dv">2</span>)<span class="op">;</span> ax.set_ylim(<span class="op">-</span><span class="dv">2</span>, <span class="dv">2</span>)<span class="op">;</span> ax.set_zlim(<span class="op">-</span><span class="dv">2</span>, <span class="dv">2</span>)</a>
    1413. 

  1413. <a class="sourceLine" id="cb82-20" title="20"><span class="co"># this next line doesn&#39;t work in colab for some reason</span></a>
    1414. 

  1414. <a class="sourceLine" id="cb82-21" title="21"><span class="co"># ax.set_box_aspect((np.ptp(x), np.ptp(y), np.ptp(x)))</span></a>
    1415. 

  1415. <a class="sourceLine" id="cb82-22" title="22">ax.tick_params(axis<span class="op">=</span><span class="st">&#39;both&#39;</span>, which<span class="op">=</span><span class="st">&#39;major&#39;</span>, labelsize<span class="op">=</span><span class="dv">6</span>)</a>
    1416. 

  1416. <a class="sourceLine" id="cb82-23" title="23"></a>
    1417. 

  1417. <a class="sourceLine" id="cb82-24" title="24">plt.show()<span class="op">;</span></a></code></pre></div>
    1418. 

  1418. <figure>
    1419. 

  1419. <img src="Vectors_And_Spaces_files/Vectors_And_Spaces_115_0.png" alt="png" /><figcaption>png</figcaption>
    1420. 

  1420. </figure>
    1421. 

  1421. <p>Let’s do another simple and obvious equation of a plane. <br><br> <br /><span class="math display">$$\begin{bmatrix} 
    1422. 

  1422. 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1
    1423. 

  1423. \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} = 
    1424. 

  1424. \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$$</span><br /> <br><br> which yields, <br><br> <br /><span class="math display">$$ \bf A = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} $$</span><br /> <br><br> Now we create our grid of points in the <span class="math inline"><em>x</em></span> - <span class="math inline"><em>y</em></span> plane, and our equation for <span class="math inline"><em>z</em></span> simplifies to <br><br> <br /><span class="math display"><em>z</em> = (1 − 0<em>x</em> − 0<em>y</em>)</span><br /> <br><br> Let’s plot this in 3D.</p>
    1425. 

  1425. <div class="sourceCode" id="cb83"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb83-1" title="1">xyz <span class="op">=</span> []</a>
    1426. 

  1426. <a class="sourceLine" id="cb83-2" title="2"><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span><span class="dv">20</span>,<span class="dv">21</span>,<span class="dv">2</span>):</a>
    1427. 

  1427. <a class="sourceLine" id="cb83-3" title="3">    <span class="cf">for</span> j <span class="kw">in</span> <span class="bu">range</span>(<span class="op">-</span><span class="dv">20</span>,<span class="dv">21</span>,<span class="dv">2</span>):</a>
    1428. 

  1428. <a class="sourceLine" id="cb83-4" title="4">        x <span class="op">=</span> i<span class="op">/</span><span class="dv">10</span></a>
    1429. 

  1429. <a class="sourceLine" id="cb83-5" title="5">        y <span class="op">=</span> j<span class="op">/</span><span class="dv">10</span></a>
    1430. 

  1430. <a class="sourceLine" id="cb83-6" title="6">        z <span class="op">=</span> <span class="dv">1</span> <span class="op">-</span> <span class="dv">0</span><span class="op">*</span>x <span class="op">-</span> <span class="dv">0</span><span class="op">*</span>y</a>
    1431. 

  1431. <a class="sourceLine" id="cb83-7" title="7">        xyz.append([x, y, z, <span class="fl">0.1</span>])</a>
    1432. 

  1432. <a class="sourceLine" id="cb83-8" title="8"></a>
    1433. 

  1433. <a class="sourceLine" id="cb83-9" title="9">dfp <span class="op">=</span> pd.DataFrame(data<span class="op">=</span>xyz, columns<span class="op">=</span>[<span class="st">&#39;X&#39;</span>, <span class="st">&#39;Y&#39;</span>, <span class="st">&#39;Z&#39;</span>, <span class="st">&#39;size&#39;</span>])</a>
    1434. 

  1434. <a class="sourceLine" id="cb83-10" title="10"></a>
    1435. 

  1435. <a class="sourceLine" id="cb83-11" title="11">line <span class="op">=</span> [[<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>], [<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">2</span>]]</a>
    1436. 

  1436. <a class="sourceLine" id="cb83-12" title="12">dfl <span class="op">=</span> pd.DataFrame(data<span class="op">=</span>line, columns<span class="op">=</span>[<span class="st">&#39;X&#39;</span>, <span class="st">&#39;Y&#39;</span>, <span class="st">&#39;Z&#39;</span>])</a>
    1437. 

  1437. <a class="sourceLine" id="cb83-13" title="13"></a>
    1438. 

  1438. <a class="sourceLine" id="cb83-14" title="14">fig1 <span class="op">=</span> px.scatter_3d(dfp, x<span class="op">=</span><span class="st">&#39;X&#39;</span>, y<span class="op">=</span><span class="st">&#39;Y&#39;</span>, z<span class="op">=</span><span class="st">&#39;Z&#39;</span>, title<span class="op">=</span><span class="st">&quot;Title&quot;</span>, size<span class="op">=</span><span class="st">&#39;size&#39;</span>)</a>
    1439. 

  1439. <a class="sourceLine" id="cb83-15" title="15">fig2 <span class="op">=</span> px.line_3d(dfl, x<span class="op">=</span><span class="st">&#39;X&#39;</span>, y<span class="op">=</span><span class="st">&#39;Y&#39;</span>, z<span class="op">=</span><span class="st">&#39;Z&#39;</span>)</a>
    1440. 

  1440. <a class="sourceLine" id="cb83-16" title="16">fig3 <span class="op">=</span> go.Figure(data <span class="op">=</span> fig1.data <span class="op">+</span> fig2.data)</a>
    1441. 

  1441. <a class="sourceLine" id="cb83-17" title="17">fig3.show()</a></code></pre></div>
    1442. 

  1442. <html>
    1443. 

  1443. <head>
    1444. 

  1444. <meta charset="utf-8" />
    1445. 

  1445. </head>
    1446. 

  1446. <body>
    1447. 

  1447. <div>
    1448. 

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    1449. 

  1449.     &lt;script src=&quot;https://cdn.plot.ly/plotly-2.8.3.min.js&quot;&gt;&lt;/script&gt;                &lt;div id=&quot;88fd770c-3552-4e86-b28e-572c93c6a97d&quot; class=&quot;plotly-graph-div&quot; style=&quot;height:525px; width:100%;&quot;&gt;&lt;/div&gt;            &lt;script type=&quot;text/javascript&quot;&gt;                                    window.PLOTLYENV=window.PLOTLYENV || {};                                    if (document.getElementById(&quot;88fd770c-3552-4e86-b28e-572c93c6a97d&quot;)) {                    Plotly.newPlot(                        &quot;88fd770c-3552-4e86-b28e-572c93c6a97d&quot;,                        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                       {&quot;responsive&quot;: true}                    ).then(function(){</code></pre>
    1450. 

  1450. <p>var gd = document.getElementById(‘88fd770c-3552-4e86-b28e-572c93c6a97d’); var x = new MutationObserver(function (mutations, observer) {{ var display = window.getComputedStyle(gd).display; if (!display || display === ‘none’) {{ console.log([gd, ‘removed!’]); Plotly.purge(gd); observer.disconnect(); }} }});</p>
    1451. 

  1451. <p>// Listen for the removal of the full notebook cells var notebookContainer = gd.closest(‘#notebook-container’); if (notebookContainer) {{ x.observe(notebookContainer, {childList: true}); }}</p>
    1452. 

  1452. <p>// Listen for the clearing of the current output cell var outputEl = gd.closest(‘.output’); if (outputEl) {{ x.observe(outputEl, {childList: true}); }}</p>
    1453. 

  1453. <pre><code>                    })                };                            &lt;/script&gt;        &lt;/div&gt;</code></pre>
    1454. 

  1454. </body>
    1455. 

  1455. </html>
    1456. 

  1456. ##
    1457. 

  1457. <center>
    1458. 

  1458. The 3D Cross Product - A Visual Method
    1459. 

  1459. </center>
    1460. 

  1460. <p><br /><span class="math display">$$ \vec a = \vec x \times \vec y = \begin{matrix} 
    1461. 

  1461.     &amp;     &amp; - x_3 y_2 \hat i  &amp; - x_1 y_3 \hat j &amp; - x_2 y_1 \hat k \\
    1462. 

  1462. x_1 &amp; y_1 &amp; \hat i &amp; x_1      &amp; y_1 \\ 
    1463. 

  1463. x_2 &amp; y_2 &amp; \hat j &amp; x_2      &amp; y_2 \\ 
    1464. 

  1464. x_3 &amp; y_3 &amp; \hat k &amp; x_3      &amp; y_3 \\
    1465. 

  1465.     &amp;     &amp; + x_1 y_2 \hat k  &amp; + x_3 y_1 \hat j &amp; + x_2 y_3 \hat i \\
    1466. 

  1466. \end{matrix} $$</span><br /> <br> <br /><span class="math display"><em>a⃗</em> = <em>x⃗</em> × <em>y⃗</em> = (<em>x</em><sub>2</sub><em>y</em><sub>3</sub> − <em>x</em><sub>3</sub><em>y</em><sub>2</sub>)<em>î</em> + (<em>x</em><sub>3</sub><em>y</em><sub>1</sub> − <em>x</em><sub>1</sub><em>y</em><sub>3</sub>)<em>ĵ</em> + (<em>x</em><sub>1</sub><em>y</em><sub>2</sub> − <em>x</em><sub>2</sub><em>y</em><sub>1</sub>)<em>k̂</em></span><br /></p>
    1467. 

  1467. <h2 id="a-simplistic-summary-of-dot-and-cross-products">A Simplistic Summary of Dot and Cross Products</h2>
    1468. 

  1468. <p><br /><span class="math display"><em>x</em> ⋅ <em>y</em> = ‖<em>x</em>‖‖<em>y</em>‖<em>c</em><em>o</em><em>s</em>(<em>θ</em>)</span><br /> <br /><span class="math display">‖<em>x</em> × <em>y</em>‖ = ‖<em>x</em>‖‖<em>y</em>‖<em>s</em><em>i</em><em>n</em>(<em>θ</em>)</span><br /></p>
    1469. 

  1469. <h1 id="stuff-for-later">Stuff For Later</h1>
    1470. 

  1470. <h2 id="a-3d-example">A 3D Example</h2>
    1471. 

  1471. <div class="sourceCode" id="cb86"><pre class="sourceCode python"><code class="sourceCode python"><a class="sourceLine" id="cb86-1" title="1"><span class="im">import</span> plotly.express <span class="im">as</span> px</a>
    1472. 

  1472. <a class="sourceLine" id="cb86-2" title="2"></a>
    1473. 

  1473. <a class="sourceLine" id="cb86-3" title="3"><span class="co"># Some New Stuff For Arrows</span></a>
    1474. 

  1474. <a class="sourceLine" id="cb86-4" title="4"></a>
    1475. 

  1475. <a class="sourceLine" id="cb86-5" title="5">fig <span class="op">=</span> px.scatter_3d(fake_df, x<span class="op">=</span><span class="st">&#39;X1&#39;</span>, y<span class="op">=</span><span class="st">&#39;X2&#39;</span>, z<span class="op">=</span><span class="st">&#39;Y&#39;</span>)</a>
    1476. 

  1476. <a class="sourceLine" id="cb86-6" title="6">fig.show()</a></code></pre></div>
    1477. 

  1477. </body>
    1478. 

  1478. </html>
    1479.
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