midterm_1_sample

midterm_1_sample - A ). c. Does there exist a vector ~x R 2...

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MA 35100 GOINS SAMPLE MIDTERM EXAM #1 This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed, but you may use your own personal notes; the lecture notes posted on the course web site; homework solutions posted on the course web site; and Bretscher’s Linear Algebra with Applications . Write your answers on a separate sheet of paper. Provide all details of your work : either give precise references for or proofs of any statements you claim. Problem 1. Consider a linear transformation T : R 2 R 3 which satisfies T ± 1 0 ² = 1 2 0 and T ± 0 1 ² = 3 0 1 . a. Find the matrix A of this linear transformation. b. Compute its reduced row-echelon form E = rref(
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Unformatted text preview: A ). c. Does there exist a vector ~x R 2 such that T ( ~x ) = 12 4 3 ? Explain. Problem 2. Consider a line L in the coordinate plane, running through the origin. The reection about L is a linear transformation in the form ref L ( ~x ) = a b b-a ~x where a 2 + b 2 = 1. What is the composition ref L ref L ? Give a geometric interpretation. Problem 3. Consider a line L in the form y = mx . a. Find a unit vector ~u which is parallel to L . (Express its entries in terms of m .) b. Find the matrix of the projection proj L ( ~x ) of ~x onto L . (Express its entries in terms of m .) 1...
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