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mat22a-practiceexam1-sol

# mat22a-practiceexam1-sol - Math 22A UC Davis Winter 2011...

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Math 22A UC Davis, Winter 2011 Prof. Dan Romik Solutions to practice question set 1 1. Find the general form of the solution of the linear system x 3 - 2 x 4 = - 2 x 1 + 4 x 2 + x 3 + 7 x 4 = 1 2 x 1 + 8 x 2 + 3 x 3 + 12 x 4 = 0 x 1 + 4 x 2 - x 3 + 11 x 4 = 5 Solution. First, write the system as an augmented matrix 0 0 1 - 2 - 2 1 4 1 7 1 2 8 3 12 0 1 4 - 1 11 5 Next, perform the Gaussian elimination procedure to bring the aug- mented matrix to reduced row echelon form (RREF) by applying a sequence of elementary row operations: 0 0 1 - 2 - 2 1 4 1 7 1 2 8 3 12 0 1 4 - 1 11 5 swap R 1 ,R 2 ------→ 1 4 1 7 1 0 0 1 - 2 - 2 2 8 3 12 0 1 4 - 1 11 5 R 3 - 2 R 1 R 4 - R 1 ------→ 1 4 1 7 1 0 0 1 - 2 - 2 0 0 1 - 2 - 2 0 0 - 2 4 4 R 1 - R 2 R 3 - R 2 R 4 +2 R 2 ------→ 1 4 0 9 3 0 0 1 - 2 - 2 0 0 0 0 0 0 0 0 0 0 Translating this back into a system of equations, we have reached the equivalent system x 1 + 4 x 2 + 9 x 4 = 3 x 3 - 2 x 4 = - 2 1

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Here we have two independent (non-pivot) variables x 2 and x 4 , so the solution can be written as x 2 = s, x 4 = t, x 1 = 3 - 4 s - 9 t, x 3 = - 2 + 2 t, where s, t are parameters taking arbitrary real values. In vector nota-
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