Matrix positive definite if and only if the vectors a lineary independent

Question

Be $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be $V$ and euclidean or unitary $\mathbb{K}$-vector space and be $v_1,\ldots,v_n \in V$.

For the $n\times n$ matrix

$$A:=(\langle v_j,v_k\rangle)_{j,k=1,\ldots,n}$$

then applys $A=A^*$ and $A\succeq0$.

Then $A$ is positive definite if and only if the vectors $v_1,\ldots,v_n$ are lineary independent.

I just can't seem to find a proof for this, and can't really proof it myself. Thanks in advance for any kind of help.

edit: the linked question doesn't talk about positive definity, or does it? Sorry if it does.

Answer 1

Hint: take the definitions of positive-definiteness and linear independence and see what you can get from $$ x^*Ax=\sum_{ij}\langle v_i,v_j\rangle \bar x_i x_j=\sum_{ij}\langle x_iv_i,x_jv_j\rangle=\left\langle \sum_i x_iv_i,\sum_j x_jv_j\right\rangle=\left\|\sum_i x_iv_i\right\|^2. $$