If a problem asks you to justify your answer, write few words explaining your conclusion.

N 6. Determine if -2 5 forms a basis for R' . Justify your answer. 2 -2 -2 0 -4 5 7. Let A = ( 0 0 0 0 -2 0 -3 5 a. Find a basis for Col A and find Rank A. b. Find a basis for Nul A. 8. Use coordinate vectors to determine if the set { 1, 1 + t, t + 2, t2 + 1 ) forms a basis for P3. Justify your answer. 9. Let B = {b1, b2} and C = {C1, C2) be bases for R2 , where b, = , b2 = ) . c -[i].e -[7']. a. Find P B+C b. If [xle = 16 find [x B5 3 10. Find the characteristic equation and eigenvalues of A = [ 1 3 ]. 11. Prove % of the following: a. Let A and B be square matrices. If det(AB) 7t 0, explain Why B is invertible. b. Suppose A is an n X 11 matrix and the equation Ax = 0 has a nontrivial solution. Explain Why Rank A