final_sample_solutions

# final_sample_solutions - MA 35100 SAMPLE FINAL EXAM...

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MA 35100 SAMPLE FINAL EXAM SOLUTIONS Problem 1. Use Gauss-Jordan elimination to solve the linear system x + y + z = 3 2 x + 3 y + 4 z = 11 4 x + 9 y + 16 z = 41 Solution: Denote the augmented matrix A = 1 1 1 3 2 3 4 11 4 9 16 41 . We will compute the reduced row-echelon form for A . Subtract twice the first equation from the second, and subtract 4 times the first equation from the third: 1 1 1 3 0 1 2 5 0 5 12 29 Now subtract the second row from the first, and subtract 5 times the second row from the third. Divide the resulting row by 2: 1 0 - 1 - 2 0 1 2 5 0 0 1 2 Finally, add the third row to the first, and subtract twice the third row from the second: rref( A ) = 1 0 0 0 0 1 0 1 0 0 1 2 We conclude that the solution is x = 0, y = 1, and z = 2. Problem 2. Fix two scalars a and b such that a 2 + b 2 = 1. Consider the identity a b b - a = a - b b a 1 0 0 - 1 . What is the geometric interpretation of this composition? Draw pictures giving a geometric interpretation of each transformation. How are reflections and rotations in the coordinate plane related to each other?

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