linear_algebra

# linear_algebra - A Primer on Solving Systems of Linear...

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A Primer on Solving Systems of Linear± Equations± In Signals and Systems, as well as other subjects in Uniﬁed, it will often be necessary to solve systems of linear equations, such as x +2 y +3 z =1 2 x +5 =2 (1) + =3 There are are at least three ways to solve this set of equations: Elimination of vari- ables, Gaussian reduction, and Cramer’s rule. These three approaches are discussed below. Elimination of Variables Elimination of variables is the method you learned in high school. In the example, you ﬁrst eliminate x from the second two equations, by subtracting twice the ﬁrst equation from the second, and subtracting the ﬁrst equation from the third. The three equations then become x 4 =0 (2) 2 =2 Next, y is eliminated from the third equation, by adding the (new) second equation to the third, yielding x 4 (3) 6 From the third equation, we conclude that 1 z = (4) 3 From the second equation, we conclude that 4 y =4 = (5) 3 Finally, from the ﬁrst equation, we ﬁnd that 14 x 2 3 (6) 3 1

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± ² ³ ´ µ ± ² ³ ´ µ ± ² ³ ´ µ ± ² ³ ´ µ ± ² ³ ´ µ ± ² ³ ´ µ Gaussian Reduction A more organized way of solving the system of equations is Gaussian reduction . First, the augmented matrix of the system is formed: 123 1 252 2 (7) 111 3 Each row of the augmented matrix corresponds to one equation in the system of equations. The ﬁrst three elements of each row are the coeﬃcients of x , y , and z in the equation.
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## This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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linear_algebra - A Primer on Solving Systems of Linear...

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