Problem 2. (9 points) If then A(ZE 33) = Are the following statements helpful? ? V Proposition 3.19 ? V Proposition 3.35 ? V Theorem 5.22 ? V Theorem 8.18 Proposition 3.19. Let A, B, and C' be m x n matrices, and let k and p be real numbers. The following equalities hold: (1) A + B = B+A (2) (A + B) +C = A+(B +C) (3) 1A = A (4) k(A + B) = KA + kB (5) (k +p) A = KA + pA (6) (kp) A = k(PA) (7) KA = AkTheorem 5.22. If the set {'31, . . . , 13%} is linearly independent, then for at! if in span', . . . ,k), there exists a unique representation of if as a linear combination of 5'1, . . . ,ic. Indeed, if for some vector if '5": (11171 + +ak'k and '17: (31171 + +bk'l7k then, equivalently, .> ((11bl)1?1+"'+(akbk)k=171?=0 Theorem 8.18. Let X1, X2, ..., Xx be eigenvectors corresponding to distinct eigenvalues 1, 12, ..., Ak of anxn matrix A. Then the vectors X1, X2, . .., Xk are linearly independent. We can try and find a basis for R" which consists of eigenvectors of a matrix A, by taking bases for distinct eigenspaces of A