Problem concerning inner product of a linear transformation and matrix transpose

Question

Let $L: \mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear map with standard matrix $A$ and let $\left \langle , \right \rangle$ denote the standard inner product on $\mathbb{R}^k$. If the matrix $B$ satisfies the relation $$\left \langle A\mathbf{v},\mathbf{w} \right \rangle = \left \langle \mathbf{v} , B\mathbf{w} \right \rangle$$ for all vectors $\mathbf{v} \in \mathbb{R}^n$ and $\mathbf{w} \in \mathbb{R}^k$, show that $B = A^T$.

I don't know where to start on this. Any help?

Answer 1

Hint:

You just need to know that the standard scalar product can be written using matrix multiplication:

• $\left \langle \mathbf{v} , B\mathbf{w} \right \rangle = v^T (Bw)$
• Now, use associativity of matrix multiplication and $(M^T)^T= M$ and $(MN)^T = N^TM^T$: $$v^T (Bw) = (v^TB)w = (v^T(B^T)^T)w = (B^Tv)^Tw = \left \langle \mathbf{B^Tv} , \mathbf{w} \right \rangle$$