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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Linear Algebra</title>
<link href="css/style.css" rel="stylesheet" type="text/css">
</head>
<body>
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<div class="main">
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<h1 class="Course_z_off">Linear Algebra</h1>
<dl>
<dt>Linear Equation</dt>
<dd><a class="Course_chapter_on" href="Chapter1_1.html">Linear Equation</a></dd>
<dd><a class="Course_chapter_off" href="Chapter1_2.html">Vector Equation</a></dd>
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<dl>
<dt>Matrix Algebra</dt>
<dd><a class="Course_chapter_off" href="Chapter2_1.html">Matrix Equation</a></dd>
<dd><a class="Course_chapter_off" href="Chapter2_2.html">Subspace & Dimension</a></dd>
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<dl>
<dt>Vector Space</dt>
<dd><a class="Course_chapter_off" href="Chapter3_1.html">Vector Space</a></dd>
<dd><a class="Course_chapter_off" href="Chapter3_2.html">Zero Space & Basis</a></dd>
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<dt>Eigenvalues</dt>
<dd><a class="Course_chapter_off" href="Chapter4_1.html">Eigenvectors&Eigenvalues</a></dd>
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</dl>
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<div class="Course_chapter">
<h1>Linear Equation</h1>
<hr color = "#708ACE" size=1px/>
<h2>What is linear equation?</h2>
<div class="Course_chapter_content">
<p>The equations look like <span>'a1*x1+a2*x2+... +an*xn = b'</span>,
5*x1 - 6*x2 = x1*x2 and x1 + 6*x1*x2 = 3 are not linear equations.</p>
<p>When a system is composed of one or more linear equations containing the same variables, it’s called System of linear equations.</p>
</div>
<hr color="#708ACE" size=1px/>
<h2>Solution of the system: </h2>
<div class="Course_chapter_content">
<p>substitute a set of Numbers for x1, x2... xn, all equations are equal to each other on both sides of the equation,
and there are three possible scenarios: no solution (incompatible), only solution (compatible), infinitely many solutions (compatible)</p>
<p>To find the solution of a system of equations consisting of two linear equations with two variables is equivalent to finding the intersection of two lines:</p>
<div align = "center" >
<img src="images/test_pairs.png" height="200px">
</div>
<h6>only one solution (compatible)</h6>
<div align = "center">
<img src="images/test_pairs_1.png" height="200px">
</div>
<h6>no solution (incompatible)</h6>
<div align = "center">
<img src="images/test_pairs_2.png" height="200px">
</div>
<h6>infinitely many solutions (compatible)</h6>
</div>
<hr color="#708ACE"size=1px/>
<h2>Elementary row operations</h2>
<div class="Course_chapter_content">
<p>Firt multiplexing: replace a row with the sum of multiples of itself and another row. Then substitution, replace two rows. At last, multiplication: multiply all the elements of a row by a non-zero number.</p>
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