a) Can a 3 × 4 matrix have independent columns? Independent rows? Explain.

b) If A is 4 × 3 and rank A = 2, can A have independent columns? Independent rows? Explain.

c) If A is an m × n matrix and rank A = m, show that m ≤ n.

d) Can a nonsquare matrix have its rows independent and its columns independent? Explain.

e) Can the null space of a 3 × 6 matrix have dimension 2? Explain.

f) Suppose that A is 5 × 4 and null(A) = x for some column x ≠ 0. Can dim(im A) = 2?