5.3 - Multivariable Linear Systems

# 5.3 Multivariable -

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<?xml version="1.0" encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>5.3 - Multivariable Linear Systems</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>5.3 - Multivariable Linear Systems</h1> <h2>Row-Echelon Form</h2> <p>Row-echelon form is basically a stair-step pattern with leading coefficients of one. When a system is placed into row-echelon form, back substitution is very easy. The bottom row gives the answer for z. That answer is back-substituted into the second equation and y is found. Then both y and z are substituted into the first equation and x is found.</p> <p>Example: Use back substitution to solve the following system.</p> <pre>x - y + 2z = 5 y - z = -1 z = 3</pre> <ol> <li>The bottom equation gives us that z = 3. </li> <li>Plugging z = 3 into the second equation gives y - 3 = -1 or y = 2. </li> <li>Plugging y = 2 and z = 3 into the first equation gives x - 2 + 2(3) = 5. </li> <li>Solving for x gives 1, so the solution is { ( 1, 2, 3 ) }.</li> </ol> <h2>Gaussian Elimination</h2> <p>Gaussian Elimination is named after Carl Friedrich Gauss, the German mathematician who proved the fundamental theorem of algebra. </p> <p>Two systems of equations are <em>equivalent</em> if they have the same solution set.</p> <h3>Elementary Operations</h3> <p>There are three basic operations, called elementary operations, that can be performed and that will render an equivalent system. </p> <ul> <li>Interchange two equations.</li> <li>Multiply one equation by a non-zero constant.</li> <li>Multiply an equation by a non-zero constant and add it to another equation, replacing that equation.</li> </ul> <p>The <a href="elimination.html">addition / elimination</a> technique that we discussed in section 5.2 uses these operations, they just weren't formalized. In the elimination technique, we could switch the two

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## This note was uploaded on 10/18/2011 for the course MAT 1033 taught by Professor Brown during the Spring '10 term at Valencia.

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5.3 Multivariable -

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