final_solutions - MA 35100 FINAL EXAM SOLUTIONS Problem 1....

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Unformatted text preview: MA 35100 FINAL EXAM SOLUTIONS Problem 1. Use Gauss-Jordan elimination to solve the linear system x- 2 y + 2 z = 2 x- 5 y + 6 z =- 1 4 x- 13 y + 18 z =- 5 Solution: Denote the augmented matrix A = 1- 2 2 2- 5 6- 1 4- 13 18- 5 . We will compute the reduced row-echelon form for A . Subtract twice the first row from the second, and subtract 4 times the first row from the third. Multiply the resulting second row by- 1: 1- 2 2 1- 2 1- 5 10- 5 Now add twice the second row to the first, and add 5 times the second row to the third: rref( A ) = 1 0- 2 2 0 1- 2 1 0 0 . That means we have the system of equations x- 2 z = 2 y- 2 z = 1 Hence z = t is a free variable while x and y are leading variables. That means the general solution is x y z = 2 + 2 t 1 + 2 t t = 2 1 + t 2 2 1 . Problem 2. Consider a line L in the coordinate plane, running through the origin. Denote proj L : R 2 R 2 as the orthogonal projection onto L , and T : R 2 R 2 as the linear transformation...
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This note was uploaded on 03/31/2012 for the course MA 351 taught by Professor ?? during the Spring '08 term at Purdue University-West Lafayette.

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final_solutions - MA 35100 FINAL EXAM SOLUTIONS Problem 1....

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