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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2015 Math 304 — Problem Set 9 Issued: 11.08 Due: 11.16 9.1. Determine whether the following are linear transformations from the space of differentiable functions to the space of functions. (a) L(f (x)) = xf 0 (x); (b) L(f (x)) = x2 + f (x). 9.2. Find the standard matrix representation (i.e., the representation in the standard bases) for the following linear operators: (a) The linear operator R2 −→ R2 that rotates each vector ~x in R2 by 45◦ in the clockwise direction. (b) The linear operator R2 −→ R2 that doubles the length of ~x and then rotates it by 30◦ in the counterclockwise direction. 2x3 x1 (c) The operator R3 −→ R3 given by the formula L x2 = x2 + 3x1 . 2x1 − x3 x3 9.3. Let S be the subspace of C[a, b] spanned by ex , xex , and x2 ex . Let D(f ) = f 0 be the differentiation operator on S. Find the matrix representation of D with respect to the basis {ex , xex , x2 ex }. 9.4. Let L be the linear operator on R3 defined by L(~x) = A~x, where 3 −1 −2 0 −2 A= 2 2 −1 −1 and let 1 ~v1 = 1 , 1 1 ~v2 = 2 , 0 0 ~v3 = −2 . 1 Find the matrix of L in the basis {~v1 , ~v2 , ~v3 }. 1 2 −1 2 . 9.5. Find characteristic polynomial of the matrix A = 0 1 0 2 1