(3) For each of the following sets, determine whether it is linearly independent or dependent. (a) (b) 3 7- (4) Let S = in R4. () in R4. (c) {x³x, 2x² +4, -2x³+3x²+2x+6} in P3 (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. 1 @ {(-12 1) (1 1¹). (1 3) (2 -¹2)} in M2x2(R), where M₂x2(R) 2 denotes the vector space of 2 x 2 matrices over the reals. CR4. Find all vectors in R4 that are orthogonal to 0 span(S), i.e., orthogonal to every vector in span(S). (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank (B). (c) If A has size m x n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity (A¹)

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