Thank you so much!

Q1. Let V = M2x2(C) be the vector space over the field C of 2 x 2 matrices with entries in C, and coefficients in C. let W = P3(C) be the vector space over the field C of polynomials of degree at most 3 with Consider the following bases B and C, of V and W, respectively: and C = {1 + it, 1 - it, 1 + t + 2it?, t + +3 ]. Let T : V - W be a function defined as follows: T = (a+ b+ c) +(b+ c+ d)t+ (c+ d)+2+ dt3 Q1(a). [2 points] Show that T is a linear transformation. Q1(b). [4 points] Find the matrix A of T with respect to the bases B and C. Show your work and briefly explain your process. Q1(c). [1 point] Suppose D is a mystery basis for W, and T (2 2 ) = (t). Write an expression for the vector of coordinates op (p(t)) in terms of da , the ma- trix A of T from (b), and the change-of-basis matrix MC-D. Your expression should tell us how we could use A to directly compute op (p(t)) if we were explicitly given the matrix Mo-D and the coordinate vector QB