(a) Recall that, for an m x n matrix A, TA : R" - RM is a linear map such that TA(X) = Ax. For the following matrices A, decide whether there exist distinct non-zero vectors x1 and X2 such that TA (X1 ) = TA(X2). 1 2 3 4 5 A = 6 7 89 2 1 0 3 8 5 ii. 1 2 3 4 5 6 A = 7 8 9 0 2 5 7 8 7 Justify your answers. [5 Marks](b) Show that the polynomials p1 = 1 + a: 3:232, p2 = 3 :I; + 12332, p3 = m + 3:132 form a basis of the vector space P2 of polynomials of degree at most 2 and find the coordinates of 1 + x in this basis. Show your working. [6 Marks] (c) Find all real numbers 35' such that the vectors (58719171): (13567171): (1719$71)7(171719$) do n_ot form a basis in R4. For each of the values that you find, determine the dimension of the subspace of R4 that they span. Show your working. [8 Marks] (d) Let U and W be two 2-dimensional subspaces in a 5-dimensional vector space V. Let U + W denote the set of all vectors in V of the form 11 + w where u E U and w E W. Prove that UlW is a subspace of V. Describe all dimensions that this subspace can possibly have. [6 Marks]