Question

detail steps please

1· Let L:R'→R' bedefined by L(x,y)-(x-2y,x+2y Let S- (1.-1).(0.D)be a basis for R' and let T be the natural basis for IR2 Find the matrix representing L w. r to a) S b) Sand T c T andS d) T e) Compute L(2,-1) using the definition of L and also using the matrices obtained in a), b), c)and d)

Answer 1

L: IR^2 rightarrow IR^2 defined by L(x, y) = (x - 2y, x + 2y) Let s = {(1, -1) , (0, 1)} be basis of IR^2 & T = {(1, 0) , (0, 1)} be standard basis of IR^2 (a) L(1, -1) = (3, -1) = 3(1, -1) + 2(0, 1) & L(0, 1) = (-2, 2) = -2(1, -1) + 0.(0, 1) Therefore Matrix w.r.t S = [3 2 -2 0] (a) L(1, 0) = (1, 1) = 1(1, 0) + 1(0, 1) & L(0, 1) = (-2, 2) = -2(1, 0) + 2(0, 1) Therefore Matrix w.r.t T = [1 1 -2 2] Ans (b) Now apply L on elements of S and write the image in Linear combination of elts of T i.e L(1, -1) = (3, -1) = 3(1, 0) - 1(0, 1) & L(0, 1) = (-2, 2) = -2(1, 0) + 2(0, 1) Therefore Matrix = [3 -1 -2 2] Ans \r\n (c) Now apply L on elements of T and write the image in Linear combination of elements of S