$\huge A_{n\times m}^{\intercal} \in M_{m\times n}$
"Rotating" a [[Matrix|Matrix]] on its side / flipping all row and column values.
$ \huge \begin{align}
\transpose{\mat{ 1 & 2 & 3 \\ 4 & 5 &6 \\ 7 & 8 & 9}} =
\mat{ 1 & 4 & 7 \\ 2 & 5 &8 \\ 3 & 6 & 9}
\end{align}$
>[!info]
>$\huge (AB)^\intercal = B^\intercal A^\intercal $
>[!note]
> For any square [[Matrix]] $A_{n\times n}$, $\transpose{A}$ will have the same *diagonal* values as $A$.
$
\begin{align}
U &= \op{Col}(A)\\
U^\intercal &= \op{Nul} \left( A^\intercal \right)
\end{align}
$