Question

Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\). If the matrix \(B\) satisfies the relation \(\langle A X, Y\rangle=\langle X, B Y\rangle\) for all vectors \(X \in \mathbb{R}^{n}, Y \in \mathbb{R}^{k}\), show that \(B\) is the transpose of \(A\), so \(B=A^{T}\). [This basic property of the transpose,
\[
\langle A X, Y\rangle=\left\langle X, A^{T} Y\right\rangle,
\]
is the only reason the transpose is important.]