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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.]
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