Let \(A\in M_{m\times n}(F)\) be a matrix with \(m\) rows and \(n\) columns and the elements \(\alpha_{ij}\), \(i=1,\ldots,m\), \(j=1,\ldots,n\).
\[A=\pmatrix{ \alpha_{11} & \alpha_{12} & \alpha_{13} & \cdots & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \cdots & \alpha_{2n} \\ \alpha_{31} & \alpha_{32} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \alpha_{n-1,n} \\ \alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{m,n-1} & \alpha_{mn}}\]
The transposed matrix \(A^T\in M_{n\times m}(F)\) is a matrix with \(n\) rows and \(m\) columns with the elements \(\alpha_{ji}\), \(i=1,\ldots,m\), \(j=1,\ldots,n\):
\[A^T=\pmatrix{ \alpha_{11} & \alpha_{21} & \alpha_{31} & \cdots & \alpha_{m1} \\ \alpha_{12} & \alpha_{22} & \alpha_{32} & \cdots & \alpha_{m2} \\ \alpha_{13} & \alpha_{23} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \alpha_{m,n-1} \\ \alpha_{1n} & \alpha_{2n} & \ldots & \alpha_{n-1,n} & \alpha_{mn}}\]
Definitions: 1 2
Explanations: 3
