ubspace of V 8. Determine whether the following are linear transformation in P, the (a) L(p(x)) = x +p(x) for p e P. (b) L(p(x)) x2p(x) p(x) for peP. 9. (a) Show that L(f(x)) = f(x) + f(0) is a linear...

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Extracted text: ubspace of V 8. Determine whether the following are linear transformation in P, the (a) L(p(x)) = x +p(x) for p e P. (b) L(p(x)) x2p(x) p(x) for peP. 9. (a) Show that L(f(x)) = f(x) + f(0) is a linear operator in C[-1,1 (b) Find kerL above. Find the range of L above. 10. Let S {(x1, 2, 3 , X4)| 31 + x2 = 3 + x4} be a subspace of R4. 11. Given v (1,-1, 1, 1) and w (4,2,2,1). (a) Determine the angle between v and w. b Find the orthogonal complement of V = span {v, w}. 12. Let A be an m x n matrix. (a) Suppose that rank A = r, what are dimensions of N(A) and b Verify that N(AT A) N(A) and rank(AT A) r.