Let $V$ be a $K$-vector space of finite dimension and let $B=\{v_1,v_2,...,v_n\}$ be a base of $V$. Let $a_{ij}=\left<v_j,v_i \right>$ be for $i,j=1,...,n$. Prove that $A=(a_{ij})$ is a positive matrix.
My attempt;
I know a positive matrix is a matriz that satisfy $A=A^*$ and $\sum_{i,j}\alpha_i\overline{\alpha_j} a_{ji}>0$ for non-zero scalars $\alpha_i, i=1,...,n$.
I think is easy to show that $A=A^*$ because $A^*=(\overline{a_{ji}})=(\overline{\left<v_i,v_j \right>})=(\left<v_j,v_i \right>)=(a_{ij})=A$
But I'm having problems to show that $\sum_{i,j}\alpha_i\overline{\alpha_j} a_{ji}=\sum_{i,j}\left<\alpha_i v_i,\alpha_j v_j \right>>0$
What can I do then? Thanks!