midterm_1_sample_solutions

midterm_1_sample_solutions - MA 35100 SAMPLE MIDTERM EXAM...

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Unformatted text preview: MA 35100 SAMPLE MIDTERM EXAM #1 SOLUTIONS Problem 1. Consider a linear transformation T : R 2 → R 3 which satisfies T 1 = 1 2 and T 1 = 3 1 . a. Find the matrix A of this linear transformation. b. Compute its reduced row-echelon form E = rref( A ). c. Does there exist a vector ~x ∈ R 2 such that T ( ~x ) = 12 4 3 ? Explain. Solution: (a) We use Theorem 2.1.2 on page 47 of the text. The matrix of the linear transformation is A = T ( ~e 1 ) T ( ~e 2 ) = 1 3 2 0 0 1 (b) To compute the reduced row-echelon form, we subtract 2 times the first row from the second, then divide the second row by- 6: 1 3 0 1 0 1 Now subtract 3 times the second row from the first, then subtract the second row from the third: E = 1 0 0 1 0 0 (c) For there to exist such a vector, we consider the matrix product T ( ~x ) = A~x i.e., 12 4 3 = 1 3 2 0 0 1 x 1 x 2 = x 1 + 3 x 2 2 x 1...
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This note was uploaded on 10/18/2010 for the course MATH 351 taught by Professor Egoins during the Spring '10 term at Purdue.

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midterm_1_sample_solutions - MA 35100 SAMPLE MIDTERM EXAM...

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