Let A, B, and M be matrices and consider the linear transformation T: R^n -> R^m defined by T(x) = Ax, where x is a vector in R^n.
a) (2 marks) Compute Tr(B), Tr(AB), and Tr(A^2B) using the given matrices. Show your work. (Do not use part a))
b) (10 marks) Find a basis for Ker(T) and a basis for Image(T). Justify your answer.
c) (2 marks) What are the rank(T) and nullity(T)? Justify your answer.
d) (6 marks) Find the matrix A. Justify your answer. (Hint: Let A = [a_ij] and compute Tr(AE_i) for each standard basis vector E_i; E_i ∈ R^n.)