How to prove that the inner space has this property?

Question

Let $A=[a_{i,j}]_{n\times n}$ be a real positive semi-definite type $n\times n$ matrix whose columns are $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_n$. Let those columns span $V \subset R^n$. How can I prove that the inner product on $V$ defined as $$<\mathbf{u},\mathbf{v}>=\mathbf{v}^TA\mathbf{u}$$ satisfies $$<\mathbf{a}_i, \mathbf{a}_j>=a_{i,j}$$?