Given a vector space with two inner products, there is a linear transformation taking one to another

Question

I am looking for some hint to the following question:

Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. Prove that there exists a linear mapping $L : V → V $such that $$[L(x), L(y)] = \langle x,y\rangle$$ for all $x,y \in V$.

Thank you.

Answer 1

Hint Pick bases $(e_a)$ and $(f_a)$ (respectively) orthonormal w.r.t $\langle \,\cdot\, , \,\cdot\, \rangle$ and $[ \,\cdot\, , \,\cdot\,]$.