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linearalgebra.html
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<?xml version="1.0" encoding="utf-8"?>
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</p>
<div id="outline-container-org88d86a2" class="outline-2">
<h2 id="org88d86a2">Linear Algebra: Systems of Linear Equations</h2>
<div class="outline-text-2" id="text-org88d86a2">
</div>
<div id="outline-container-org1d525cc" class="outline-3">
<h3 id="org1d525cc">Systems of linear equations</h3>
<div class="outline-text-3" id="text-org1d525cc">
<p>
<font color = "#650d1c"> ➝ An equation in the unknowns \(x\;\),
\(y\;\), \(z\;\), …, is called <i>linear</i> if both sides of the equation
are a <i>sum</i> of (constant) multiples of \(x\;\), \(y\;\), \(z\;\), …, plus
an optional constant<label for="1" class="margin-toggle sidenote-number"></label><input type="checkbox" id="1" class="margin-toggle"/><span class="sidenote">
We will usually move the unknowns to the left side of the
equation, and move the constants to the right.
</span>. </font>
</p>
<ul class="org-ul">
<li>A <i>solution</i> of a system of equations is a list of numbers \(x\;\),
\(y\;\), \(z\;\), … that make all of the equations true
simultaneously.</li>
<li>The <i>solution set</i> of a system of equations is the collection of all
solutions.</li>
<li><i>Solving</i> the system means finding all solutions with formulas
involving some number of parameters.</li>
</ul>
<p>
<font color = "#650d1c">
➝ A system of equations is called <i>inconsistent</i> if it has no solutions. It
is called <i>consistent</i> otherwise.
</font>
</p>
<p>
<font color = "#650d1c"> ➝ Let \(n\;\) be a positive whole
number. We define \(\mathbb{R^n}\;\) = all ordered n-tuples of real
numbers \((x_1, x_2, x_3,\ldots, x_n )\;\;\;\). </font>
</p>
<p>
Hence, an <i>n</i>-tuple of real numbers is called a <i>point</i> of
\(\mathbb{R^n}\;\;\)
</p>
<p>
𝖟𝕭: The real number line is when \(n\;= 1\;\) of \(\mathbb{R^n}\;\).
</p>
<p>
𝖟𝕭: Consider the linear equation \(x + y + z = 1\;\;\). This is the
implicit equation for a plane in space that is hinged on the three
coordinates totaling \(1\;\;\).
</p>
<img src="./images/planein3Dspace.png" width="300px" style="padding: 15px 0px 0px 0px" alt="3DPlane" class="center">
<span class="cap">The plane defined by x + y + z = 1</span>
<img src="./images/xplusypluszequals1.png" width="400px" style="padding: 15px 0px 0px 0px" alt="3DPlane" class="center">
<span class="cap">Another view of the plane defined by x + y + z = 1</span>
<pre class="code"><code>
</code></pre>
</div>
</div>
<div id="outline-container-org7fa81a8" class="outline-3">
<h3 id="org7fa81a8">Examples</h3>
<div class="outline-text-3" id="text-org7fa81a8">
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">a</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>4<span class="org-haskell-operator">><</span>3<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span> 1, 2, 3
, 4, 0, 5
, 7, 7, 2
, 3, 3, 1<span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">b</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>3<span class="org-haskell-operator">><</span>1<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span> 2
, 2
, 2<span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-definition">a</span> <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> b
</code></pre>
<pre class="example">
(4><1)
[ 12.0
, 18.0
, 32.0
, 14.0 ]
</pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">c</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>2<span class="org-haskell-operator">><</span>3<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span> 4,5,6
, 7,8,9<span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">d</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>3<span class="org-haskell-operator">><</span>1<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span> 1
, 1
, 1<span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">e</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>3<span class="org-haskell-operator">><</span>2<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span>1<span class="org-haskell-operator">..</span><span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-definition">c</span> <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> d
</code></pre>
<pre class="example">
(2><1)
[ 15.0
, 24.0 ]
</pre>
<pre class="code"><code><span class="org-haskell-definition">a</span> <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> e
</code></pre>
<pre class="example">
(4><2)
[ 22.0, 28.0
, 29.0, 38.0
, 38.0, 54.0
, 17.0, 24.0 ]
</pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">f</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>3<span class="org-haskell-operator">><</span>2<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span>1<span class="org-haskell-operator">..</span><span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code><span class="org-haskell-constructor">:</span><span class="org-rainbow-delimiters-depth-1">{</span>
<span class="org-haskell-definition">g</span> <span class="org-haskell-operator">=</span> <span class="org-rainbow-delimiters-depth-2">(</span>3<span class="org-haskell-operator">><</span>2<span class="org-rainbow-delimiters-depth-2">)</span>
<span class="org-rainbow-delimiters-depth-2">[</span>6,5<span class="org-haskell-operator">..</span><span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-haskell-operator">::</span> <span class="org-haskell-type">Matrix</span> <span class="org-haskell-type">R</span>
<span class="org-haskell-type">:</span><span class="org-rainbow-delimiters-depth-1">}</span>
</code></pre>
<pre class="code"><code>f <span class="org-haskell-definition">+</span> g
</code></pre>
<pre class="example">
(3><2)
[ 7.0, 7.0
, 7.0, 7.0
, 7.0, 7.0 ]
</pre>
\begin{align*}
A(u + v) = Au + Av
\end{align*}
<pre class="code"><code><span class="org-rainbow-delimiters-depth-1">(</span>a <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> <span class="org-rainbow-delimiters-depth-2">(</span>f <span class="org-haskell-operator">+</span> g<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-haskell-operator">==</span> <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>a <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> f<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-haskell-operator">+</span> <span class="org-rainbow-delimiters-depth-2">(</span>a <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> g<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>
</code></pre>
<pre class="example">
True
</pre>
\begin{align*}
A(cu) = cAu
\end{align*}
<pre class="code"><code><span class="org-haskell-definition">a</span> <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> <span class="org-rainbow-delimiters-depth-1">(</span>5 <span class="org-haskell-operator">*</span> f<span class="org-rainbow-delimiters-depth-1">)</span>
</code></pre>
<pre class="example">
(4><2)
[ 110.0, 140.0
, 145.0, 190.0
, 190.0, 270.0
, 85.0, 120.0 ]
</pre>
<pre class="code"><code><span class="org-rainbow-delimiters-depth-1">(</span>a <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> <span class="org-rainbow-delimiters-depth-2">(</span>5 <span class="org-haskell-operator">*</span> f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-haskell-operator">==</span> 5 <span class="org-haskell-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>a <span class="org-haskell-operator">Numeric.LinearAlgebra.<></span> f<span class="org-rainbow-delimiters-depth-1">)</span>
</code></pre>
<pre class="example">
True
</pre>
<p>
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<!-- Footnotes --><!--
<div class="footdef"><sup><a id="fn.1" class="footnum" href="#fnr.1" role="doc-backlink">1</a></sup> <div class="footpara" role="doc-footnote"><p class="footpara">
We will usually move the unknowns to the left side of the
equation, and move the constants to the right.
</p></div></div>
--></div>
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