Question 1 (21 marks) (a) Let A = (dij)nxn be an n x n real matrix. (i) Give the o-E definition of det (A) . Be sure to define all the non- obvious symbols used. (3) ii) Use the definition in (i) to prove that if B is obtained from A by the elementary row operation Rm - Rm + BR, then det (B) = det (A) . (3] (b) Consider the matrix B = By considering the solutions to the equation Bx = 0 or otherwise, (i) prove that the null space or the solution space of B, Null(B) , is a subspace of R4 [4] (ii) find, Null(B) [3] (iii) find a basis for Null(B ) . [2] (c) Let V be the 4 dimensional vector space ?*2, the set of all 2 x 2 matrices with real entries. (i) Show that the subset A = 1 ( 2 3) ( 2 ) (12 )} of V is linearly dependent. [4] (ii) Hence, express the first vector in the set in terms of the other two vectors of the set. [2]Question 2 (20 marks) (a) Let Vi and V2 be finite dimensional vector spaces. You are given that T : VI - V2 is a linear bijection (injective and surjective). (i) What is meant be the statement that T : V] - V2 is a linear transformation? (ii) What is meant be the statement that T : V, - V2 is (a) injective (b) surjective? (3) (b) Prove that if T : V, - V2 is a linear bijection and that if B1 = (v1 , U2, . . . . Un} is a basis for Vi then B2 = IT (u, ) , T (v2) , . ...T (u.) } (i) spans V2 [3] (ii) is linearly independent (3] (iii) is a basis for V2 [1] (c) Let Pr be the vector space of all polynomials of degree n or less. A linear transformation T is defined by T : P4 - P2 such that Tip (t)] = p" (t) where p" (t) is the second derivative of p with respect to t. (i) What is T(at* + bt3 + ct2 + dt + e)? [1] (ii) With respect to the ordered standard bases, determine the trans- formation matrix, A, representing T. [3] (iii) Use A to confirm T (at* + bt3 + ct2 + dt + e) found in part (i). [2] (iv) Find Ker(T) . [2]Question 3 (14 marks) (a) Let P2 (t) be the vector space of real coefficients polynomial of degree 2 or less. An ordered basis for Pz (t) is B = (1+ t?, 1 + t, t + t? ] What are the coordinates of p(t) = 8t2 + t + 3 (i) with respect to the ordered standard basis for P2 (t) [1] (ii) with respect to B. (b) The set G = {, [2, Is, LA} is orthogonal. (i) What does it mean by saying that G is orthogonal? 2 (ii) Prove that G is linearly independent. [3] (c) The set S = {(1, 2, 1, 2) , (0, 1, 0, 1) , (1, 0, 0, -1) } is a basis for W, a subspace of R4. (i) Find an orthogonal basis for W. (4] (ii) Find an orthonormal basis for W. [1