6. (10 points) Is the set S = {(1,5,3), (0,1,2), (1,6,7), (-1, -3,2)} a basis for the vector space R3? Briefly explain your answer. 7. (10 points) Let B = {(1,2,3) , (1,2,0), (0, -6,2)] and x = (1, -2,0). If B is a basis for 3, find [x]B. If B is not a basis, explain why it isn't. 8. (10 points) Given B = {(1,1, -1) , (1,1,0), (1, -1,0)} and B' = { (1,2,3) , (1,2,0), (0, -6,2)]. Find the transition matrix from B to B'. Use your result to find [X]g' for [x]B = N 9. (10 points) The matrix B has been obtained from a matrix A by row operations. 0 0 B = 0 - 0 0 OHO OH 0 0 Use the matrices to find the following: a. The rank and the nullity of A. b. The rank and nullity of AT. c. A basis for the null space of A. d. A basis for the row space of A. e. A basis for the column space of A. 10. (10 points) Consider the matrix A = [7 8 7'] a. Is in the column space of A? Justify your answer. b. Is [1 1 - 2] in the row space of A? Justify your answer. c. Find any vector in the null space of A. Justify your