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Math 304 (Spring 2015) - Homework 6 Problem 1. 1 1 Find the transition matrix from the basis {u1 = , u2 = } to the 1 −1 1 0 standard basis {e1 = , e2 = } 0 1 Problem 2. Let P2 be the vector space of polynomials with degree ≤ 2. We know that {1, x, (x + 1)2 } is a basis of P2 . Find the coordinate vector of the polynomials p(x) = x2 − 1 with respect to the basis {1, x, (x + 1)2 }. Problem 3. Given the matrix −3 1 3 4 A = 1 2 −1 −2 −3 8 4 2 (a) Find a basis of the row space of A and use it to determine the rank of A. (b) Find a basis of the column space of A. (c) Find a basis of the null space of A. Problem 4. Determine whether the following mappings are linear transformations. (a) L : R3 → R2 by (b) L : R3 → R2 by a a+b L b = c c 2 a 2 a + b L b = c c (c) Let P3 be the vector space of all polynomials with degree ≤ 3. The mapping L : P3 → P3 by L(p(x)) = p0 (x) where p0 (x) is the derivative of p(x). 1 (d) L : P2 → P3 by L(p(x)) = x · p(x) (e) L : P2 → P3 by L(p(x)) = p(x) + x2 Problem 5. Find the matrix representations of the following linear transformations. (a) Let L be the linear transformation from R3 to R3 by x1 2x1 − x2 − x3 L x2 = 2x2 − x1 − x3 x3 2x3 − x1 − x2 Find the standard matrix representation of L. (b) The vectors 1 1 0 v1 = 1 , v2 = 0 , v3 = 1 0 1 1 form a basis of R3 . Let L be the linear transformation from R2 to R3 defined by x L 1 = x1 v1 + (x2 + x1 )v2 + (x1 − x2 )v3 . x2 Find the matrix representation of L with respect to the bases {e1 , e2 } (the standard basis of R2 ) and {v1 , v2 , v3 }. (c) The vectors 0 1 1 v1 = 1 , v2 = 0 , v3 = 1 1 1 0 form a basis of R3 . Let L be the linear transformation from R2 to R3 defined by x L 1 = x1 v1 + (x2 + x1 )v2 + (x1 − x2 )v3 . x2 Find the matrix representation of L with respect to the standard bases. 2