Question

1. Let L: R2-R2 be defined by L(x.y) (x +2y, 2x - y). Let S be the natural basis of R2 and let T = {(-1,2), (2,0)) be another basis for R2 . Find the matrix representing L with respect to a) S b) S and1T c) T and S d) T e) Find the transition matrix Ps- from T basis to S basis. f) Find the transition matrix Qre-s from S-basis to T-basis. g) Verify Q is inverse of P by QP PQ I 2x2. h) Verify PA,P A

Answer 1

Given L: r^2 rightarrow r^2 be defined by L(x, y) = (x + 2 y, 2 x - y) , S is natural basis of r^2 i.e S = {(1, 0) , (0, 1) } T = {(-1, 2) (2, 0)} another basis of R^2 (a) Matrix of L with respect S Calculate l(1, 0) = (1 + 2(0), 2(1) 0) l(1, 0) = (1, 2) --(i) L(1, 0) = (1, 0) + (0,2) L(1, 0) = 1(1, 0) + 2 (0, 1) \ L(0, 1) = (0 + 2 (1) , 2(0) - 1) L(0, 1) = (2, -1) --(2) L(0, 10 = (2, 0) + (0, -1) L(0, 1) = 2(1, 0) + (-1) (0, 1) [L(1, 0)]_s = [1 2] , [L (0, 1)]_s = [2 -1] The coordinate vectors of images of elements of s under L with respect to basis 's'. Thus matrix of L with basis S is [[L(1, 0)_S [L(0, 1)]_s] = [1 2 2 -1] b) Matrix of L with respect to S and T from (1) (2)