1. Let g: 183 - R* defined by g (XXX;) = (X3 - Xux2 - X1, -X2). V(XX2 X;) ER3. (a) Show that g is a linear transformation. [4 marks] (b) Consider the ordered basis B = {(0,0,1), (0,2,0), (-1,0,0)]. (i) Find the matrix [gle of g with respect to B. 15 marks] (ii) Suppose that y 6 13 such that [u] = (2, -1,2)". Use the matrix obtained in 1(b)(1) to compute T (70@)), [4 marks] 2. Suppose that (, ): R* x R* - R defined by (x, y) = (x2 - x1)(12 - Vi) + x2)/2 + 2x3)3. for x = (X1, X2, x3). y = (V1. V2. Va) ER3. (a) Prove that (,-) is an inner product on R3. [7 marks] (b) Let S = [(0,1,0) (0,2,1), (1,0,0)]. Given that $ is linearly independent, use the Gram-Schmidt Orthogonalization Process to obtain an orthonormal basis for R with respect to the given inner product (,). [10 marks] BONUS QUESTION (5% of Overall Mark) Let A = (a) Find the eigenvalues of A. (b) Obtain an eigenvector corresponding to each eigenvalue of A. (c) Hence, find an invertible matrix P and a diagonal matrix D such that P-1AP = D