Show that if the columns of an m times n matrix A are linear

Show that if the columns of an m times n matrix A are linearly dependent, then A^T A is not invertible. (Do NOT use Theorem 14-you are proving part of it!)

Solution

Let A be a m x n matrix with linearly dependent columns. Let us also assume that A^TA is an invertible matrix.

We know that the matrix A^T has the same rank as A ( as the column rank and row rank are the same). Since the rank of a matrix is the dimension of the image of a mapping represented by the matrix, hence, if B is an n x m matrix , then rank(AB) min(rank(A),rank(B)).Therefore, rank(A^T A) min (rank(A^T),rank(A)). Now, since A^TA is an n x n invertible matrix, its rank is equal to n. However, if the columns of A are linearly dependent, then rank(A) < n . This implies that rank(A^T A) < n, a contradiction. Hence A^T A is not invertible.