Showing that a set of vectors is linearly independent if the induced matrix of inner products between those vectors is invertible

Question

I'm working on the following problem:

Let $V$ be a finite-dimensional real vector space, and let $( , )$ be a positive definite inner product on $V$. Show that elements $v_1,\ldots,v_m \in V$ are linearly independent if and only if the $m \times m$ matrix $A = (a_{i,j})$ given by $a_{ij} = (v_i, v_j)$ is non-singular.

I have figured out the first direction, but not the second. Does anyone have any ideas?