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final_solutions

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

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MA 35100 FINAL EXAM SOLUTIONS Problem 1. Use Gauss-Jordan elimination to solve the linear system x - 2 y + 2 z = 0 2 x - 5 y + 6 z = - 1 4 x - 13 y + 18 z = - 5 Solution: Denote the augmented matrix A = 1 - 2 2 0 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 0 0 1 - 2 1 0 - 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 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 0 + 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 T ( ~x ) = 0 - 1 1 0 ~x.

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

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