I need help on those problems please!

1. Let & = {(1, 0, 0), (0, 1, 0), (0, 0, 1) } and B = {(1, 0, 0), ( 0, 1, 1), (0, 2, 1) } be two ordered bases for R3. Let T: R3 - R3 be the linear operator on R3 defined by T(x, y, z) = (-2x, 3y + 2z, -2y - z) for each (x, y, z) E R3. (a) (4 pts) Use definition to compute [Ts. Show that T is invertible and find a formula for T-1(x, y, z) for each (x, y, z) ER3. (b) (6 pts) Use definition to compute ITis. Show your work. Confirm your answer2. Let & = {Ell, E12, E21, E22) be the standard ordered basis for M2x2 (R), where M2x2(R) is the vector space of all 2 x 2 real matrices under the usual matrix addition and scalar multiplication. Let T: M2x2(R) - M2x2(R) the linear operator defined by T(B) = B, the transpose of B, for each B E M2x2 () (a) (4 pts) Use definition to compute matrix A = [T s. Determine all values ) such that det(A - XI) = 0, where I is the identity 4 x 4 matrix. Show your work. (b) (8 pts) For each eigenvalue A obtained in (a), find its corresponding eigenspace Ex = N(A - XI) = {x ER4 : (A - XI)x = 0} and determine a basis for Ex. Show your work. (c) (3 pts) Use (b) to show that there exists a basis 3 for R* such that Q-AQ is diagonal matrix for some invertible matrix Q as in Theorem 2.23 (Section 2.5 on Page 113) and the paragraph after Example 6 in Section 5.1 on Pages 251-252