Transpose of a Real Matrix

Let $m$ and $n$ be natural numbers, and let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, indexed as in the definition of the matrix-vector product: the index $\alpha\in\{1,\dots,m\}$ labels rows and the index $i\in\{1,\dots,n\}$ labels columns.

The transpose of $A$ is the $n\times m$ matrix with real entries, denoted $A^{\top}$, whose entries are defined by

for every $i\in\{1,\dots,n\}$ and every $\alpha\in\{1,\dots,m\}$.