Question

10. (10 pt) Let bi1 The set B tbi,b2) is a basis for R2. Let T : R2 →R2 is a linear transformation such that T(bi) 7bi +7b2 and T (b2) 3bi +4b2 Then the matrix of T relative to the basis B is and b2- -1 4 [T]B and the matrix of T,relative to the standard basis E for R2 is

Answer 1

Given L: R^2 rightarrow R^2 is a linear transformation. Such that T (b_1) = 7 b_1 + 7 b_2 and T (b_2) = 3 b_1 + 4 b_2 T [b_1 b_2] = [7 b_1 + 7 b_2 3 b_1 + 4 b_2] Then (T)_B = [7 3 7 4] Let E = {e_1, e_2} be the standard basis Take P = [- 1 - 1 3 4] as b_1 = [- 1 - 1] & b_2 = [3 4] Now P^- 1 = 1/ad - bc [a - c - b a] (inverse matrix of P [a c b d] = 1/4 (- 1) - 3 (- 1) [4 1 - 3 - 1] = 1/- 4 + 3 [4 1 - 3 - 1] = 1/- 1 [4 1 - 3 - 1] = [- 4 - 1 3 1] \r\n The matrix (T)_B & (I)_E related as T_E = P middot T_B middot P^- 1