Question

Prove or disprove: If the columns of B(n×p) ? R are linearly independent as well as those of A, then so are the columns of AB (for A(m×n) ? R ).

Answer 1

Let B = [b₁,b₂,…,b_p], where b_i   (1 ? i ? p) is the ith column vector of B. Further, AB = [Ab₁,Ab₂,…,Ab_p] . Let us assume that the columns of a mx n matrix A and a nxp matrix B with real entries are linearly independent while the columns of AB are linearly dependent. Then there exist real scalars x_i (1 ? i ? p), not all zero such that x₁Ab₁+x₂ Ab₂ +…+x_p Ab_p = 0. Then A(x₁b₁ +x₂b₂+…+x_pb_p) = 0 so that x₁b₁ +x₂b₂+…+x_pb_p = 0. However, since the columns of B are linearly independent, this means that all the x_i s are 0 which is a contradiction. Hence the columns of AB are also linearly independent.