If the visual manifestation of a matrix is a linear transformation, what is that of its transpose?
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It corresponds to the adjoint of that linear transformation. If $A: X \rightarrow Y$ is a linear map between two (real) vector spaces, then $A^T: Y^* \rightarrow X^*$ is defined by $\langle Ax, y \rangle = \langle x, A^T y \rangle$ for all $x \in X$. – Matt Mar 13 '17 at 12:24
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@Widawensen The duplicates have many answers, also about "visual manifestations" in the cited duplicate. – Dietrich Burde Mar 13 '17 at 13:53