PART 1. TnuEfFAIJii-z :9 problems will be graded at random ; 1 pt. ea.] 1. Mark each statement True or False. Justify each answec [If true, cite appropriate facts or theorems or prove the statement. If false, explain why or give a counterexample that shows why the statement is not true in every case] For example. consider the following statements about it x n. matrices A and B: [it A + B = B + A [ii] AB = BA Statement ii] is always true. Explanation: The (i. 3'} entry of A + B is [1,-3- + bt-J- and the (i, 3'} entry of B + A is bij + at- -. Since oi].- + bi]- = b + at} for each 1' and it it follows that A + B = B + A. Statement {ii} is false. Explanation: Although the statement may be true in some cases, it is not always true. To show this. we need only exhibit one instance in which equality fails to hold. For example, if 1 a as A=[3 1] \""d [1 1] 4 5 11 '3" . .. . then AB [1, 1D] and [4 3].Thrs proves iatstatementlirlrsfalse. (j) If v1, V2, Va, and v4 are in RA and {V1, V2, V3, VA} is linearly dependent, then {V1, v2, Va} is linearly dependent. (k) If V = R and addition and scalar multiplication are defined as rey=r+ly com=>+c then V is a vector space. (1) The set S = {AE M2x2 | det(A) = 0} is a subspace of M2x2. (m) The set { x3 - 2x' + 1, x2 - 4, x3 + 2r, 5x} is a basis for 23. Note: P3 is the set of all polynomials of degree at most 3. (n) The dimension of the subspace of R3 is 2. (o) If v1, v2, va is a basis for a vector space V and w1 = v1 + 2v2 + V3, W2 = V1 + V2 + V3, W3= V1 - V2 - Va, then W = {w1, w2, wa} is also a basis for V. (p) In a vector space V. If span ( V1, V2, . . ., Vn) = V and w1, W2, ...; Wm are any elements of V, then span { V1, V2, . . ., Vn, W1, W2, . .., Wm) = V