MATH 225 Spring 2017: Assignment 3 Due at 8:25 a.m. on Friday, May 26, 2017 It is important that you read the assignment submission instructions and suggestions available on LEARN. \b 1. Let B = 1 + 4x2 , 2 3x 7x2 , 1 7x x2 be a basis for P2 . (You do not need to verify that B is a basis.) If p(x) = 1 11x + 9x2 , find [p(x)]B , the coordinates of p(x) with respect to B. Page 1 0 0 1 6 1 2 3 0 , 1 , 0 1 , 0 , 3 be another 2. Let S = be the standard basis for R . Let B = 0 0 1 4 0 13 basis for R3 . (You do not need to verify that B is a basis.) Find the change of coordinates matrix from B-coordinates to S-coordinates and the change of coordinates matrix from S-coordinates to B-coordinates. Page 2 3. Determine if the mapping L : P1 P2 defined by L(p(x)) = (1 + x)p(x) is linear. Justify your answer. Page 3 3 4. Find a basis for the rangeand a basis for the nullspace of the linear mapping L : P3 R defined by 0 L(a + bx + cx2 + dx3 ) = 2b . Verify that the Rank-Nullity Theorem holds true for this linear a 3d mapping. Page 4 5. Let V and W be vector spaces over R and let L : V W be a linear mapping. Prove that Null(L) is a subspace of V. Page 5 \f\f\f\f\f\f\f\f\f\f