Invertible matrix of inner product values Let $\{v_1,v_2,\ldots,v_k\}$ be a basis in inner product space $V.$
I need to prove that the matrix
$$A=
\begin{pmatrix}
        (v_1,v_1) & (v_1,v_2) & \cdots &(v_1,v_k) \\
        \
        \vdots & \vdots & \ddots&\vdots\\ 
        (v_k,v_1) & (v_k,v_2) & \cdots&(v_k,v_k)
        \end{pmatrix}
$$
is invertible.
Any hints?

Hint 1

 Take a linear combination of the rows which is zero.

Hint 2

 So there are coefficients $a_{i}$ such that $\sum_{i=1}^{k} a_{i} (v_{i}, v_{j}) = 0$ for all $j$.

Hint 3

 Your aim is to show that all $a_{i}$ are zero.

Hint 4

 So for the vector $v = \sum_{i=1}^{k} a_{i} v_{i}$ you have $(v, v_{j}) = 0$ for all $j$.

