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

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