e2-sample - n A-1 3 Find the inverse of the matrix A = 1...

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Math 204. Exam 2 (practice) October 13th, 2010 This is the practice exam. The actual exam is about the same length. Note that this is supposed to give you a feeling for what the actual exam will be like. It does not represent every “type” of problem. 1. Determine whether each of the following functions is linear. If it is prove it, if not give an example. (a) T : R n R given by T ( x ) = k x k 2 (b) T : R n R n given by T x 1 . . . x 3 = x 1 x 2 - x 3 2 x 1 + 3 x 3 - x 2 . 2. Suppose A and B are invertible matrices. Which of the following statements is true. If the statement is true, prove it. If false, then give a counterexample. (a) ( AB ) - 1 = B - 1 A - 1 . (b) ( A + B ) - 1 = A - 1 + B - 1 (c) ( ABA - 1 ) n = AB
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Unformatted text preview: n A-1 . 3. Find the inverse of the matrix A = 1 0-1 0 1 2 1 1 1 4. Find the matrix of the linear transformation that rotates R 2 through an angle of 120 ◦ and then reflects all vectors in the line y = 3 x . 5. Using the Gram-Schmidt process or otherwise, find an orthonormal basis for the hyperplane x 1 + x 2 + ...x n = 1. 6. Suppose that T : R n → R n is given by T ( ~e j ) = ~e j +1 for j = 1 ,...,n-1, T ( ~e n ) = ~e 1 . (a) Describe the matrix associated to T . (b) Find all matrices (or linear transformations) that commute with T . 1...
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