Show that if the columns of B are linearly dependent, then so are the columns of AB.

Question

Show that if the columns of B are linearly dependent, then so are the columns of AB.

Answer 1

Suppose B is an n x n matrix with linearly dependent columns B₁,...,B_n. There are constants c₁,...,c_n, not all zero, such that c₁B₁+•••+c_nB_n = 0. Pre-mulitply this equation by A to obtain c₁AB₁ +•••+ cAB_n = 0. Since the matrix AB has columns AB₁,...,AB_n, then the columns of AB are linearly dependent.