6. (9 points) Let A=(a 1))? Cd and let A = (a d)2 + Abe. Prove that A has two distinct real eigenvalues if A > 0, and A has exactly one real eigenvalue if A = 0. 7. (10 points) Let A be a 3 X 3 matrix With exactly two distinct real eigenvalues A1 and A2 such that the eigenspace EA, has basis {u, 1)} and the eigenspace EM has basis {11)}. Prove that A is diagonalizable by showing that {L5, 2), w} is a basis for R3. 8. (10 points) Let V be a vector space, let {u,1,. .. ,um} C V be linearly independent, and lot 11) E V. Prove that either 10 E span{u1, . . . ,um}, or {151, . . . ,um,w} is linearly independent