8. Determine which of the following sets are linear independent: (a) {(4, 4,8,0), (2, 2, 4, 0), (6, 0, 0, 2), (6, 3, -3,0)} in R4. (b) {2, 4 sin x, cos x} in C[-T, ]. - (c) {t3-5t2-2t+ 3, t - 4t - 3t+4,2t - 7t 7t+9} in P3, where P3 is the set of polynomials over R with degree at most 3. 9. Let f1, f2 C[-1, 1] be defined as fi(t) =t, t (-1, 1] and f2(t) = { -t if t [-1,0], t if t [0, 1]. Show that the set {f1, f2} is linearly dependent in C[0, 1] and in C[-1,0], but linearly independent in C[-1, 1]. 10. Show that the set {1+i, 1-i} C C of vectors is linearly independent if C is taken as a vector space over R. But it becomes linearly dependent when C is a vector space over C. 11. Show that u, U2,..., uk Rn are linearly independent if and only if Au, Au2,..., Auk are linearly independent for any invertible n x n matrix A. 12. Let V be a vector space over a field F. Let A and B be two non-empty subsets of V. Prove or disprove: Span(A) Span(B) AnBo, where Span(A) denotes the spanning set of A. {0}