MT2sol - Stony Brook University Introduction to Linear...

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Unformatted text preview: Stony Brook University Introduction to Linear Algebra Mathematics Department MAT 211 Julia Viro April 21, 2009 Solutions for Midterm 2 Problem 1. A linear transformation T : R 2 R 2 is defined by T ( x,y ) = (- x + 2 y, 3 x- y ) . Find a matrix of T with respect to the basis B = { (2 , 1) , (1 , 1) } in R 2 . Is T an isomorphism? Explain! Solution. A matrix of T with respect to the basis B = { (2 , 1) , (1 , 1) } is T B = | | T (2 , 1) T (1 , 1) | | , where the coordinates of the column vectors are given in the basis B . We calculate T (2 , 1) = (0 , 5) =- 5 (2 , 1) + 10 (1 , 1) T (1 , 1) = (1 , 2) =- 1 (2 , 1) + 3 (1 , 1) and get the matrix: T B =- 5- 1 10 3 . Another way to solve the problem is to consider the composition R 2 id- S B St R 2 T- T St R 2 id- S St B R 2 of T and the identity transformations of R 2 . The matrix of the composition is the product of three matrices: T B = S St B T St S B St = S- 1 T B S, where S = S B St is the transition matrix from basis...
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MT2sol - Stony Brook University Introduction to Linear...

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