I am not asking if we can build an orthogonal vector space using GS algorithm.
Let $V$ be a finite vector space with basis $B$, show that exists an inner product $b(\cdot,\cdot )$ such that $V$ is orthogonal under $b$.
At first, I thought of defining $b(x, y)=[x^T]_B[I]_B[y]_B=[x^T]_B[y]_B$, it works, but how do I prove it?
We just need to show linearity of $b$.
Let $x,y,z\in V, \alpha \in F$
$$b(\alpha x+y,z)=\left<[\alpha x+y]_B, [z]_B\right>=\left<\alpha[x]_B+[y]_B, [z]_B\right>=\left<\alpha[x]_B,[z]_B\right>+\left<[y]_B+[z]_B\right>=\alpha b(x,z)+b(y,z)$$
And then, $b$ is linear; it also follows that for any $v_i, v_j \in B$ where $i \ne j$, $\left<[v_i]_B, [v_j]_B\right>=0$