Question

ame 5. (20 points) Suppose SRR and TRR2 are linear transformations given by (o) Find the standard matrix for S (asuming the standard basis for R3 and for R2) (b) Find the standard matrix for T (assuming the standard basis for IR2 and for R). (c) Show that S and T are invertible. (d) Show that T is the inverse of S (e) What is the standard matrix for T (S (x)) ToS(),where z e R27

Answer 1

s: R^2 rightarrow r^2 by s(x, y) = (3 x - y, - x) a) s(1,0) = (3, -1) = 3(1,00 + (- 1)(0,1) s(0,1) = (- 1,0) = -191,00 + 0(0,1) where beta = {(1,0) ,(0,1)} is the standard basis for R^2 thus [s]_beta = [5 - 1 - 10] = A b)Define T: R^2 rightarrow R^2 by T(x, y) = (- y, - x - 2 y) t(1,0) = (0, - 1) = 0(1,00 + (- 1)(0,1) T(0,10 = (4, - 3) = -1(1,00 + 9- 3) (0,1) [T}_beta = [0 0 1 - 1 -3] = B c) det (A) 1 notequalto 0 and det(B) = 1 notequalto 0 since the standard material of A s and T are invertible, hence the transformations s and T are invertible d) consider A_0 = [3 - 1 - 1 0][0 - 1 - 1 3] = [1 0 0 1] = I and B_0 A = I hence A = B^-1 i e [T]_beta^-1 = [s]_beta Therefore T is inverse of s \r\n since b_o A = I hence the standard matrix of T(s(x) is the identity matrix

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