\(
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\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
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\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Algebra
Transpose characterized
Instead of thinking of the matrix transpose as the swapping of rows and
columns, there is a more meaningful characterization:
Matrix transpose
Let \( A \) be an \( m \times n \) matrix and \( x \in \mathbb{R}^n \)
and \( y \in \mathbb{R}^m \) be column vectors. Then the transpose of \( A \),
denoted \( A^{T} \) is the matrix such that the following holds:
[\[
\; ? \quad = \quad ? \;
\]]
