#math/linear-algebra Suppose $m$ and $n$ are nonnegative integers. An $m\text{-by-}n$ matrix $A$ is a rectangular array of elements of $\mathbf{F}$ with $m$ rows and $n$ columns. $ A=\begin{pmatrix} A_{1,1} & \ldots & A_{1,n} \\ \vdots{} & & \vdots{} \\ A_{m,1} & \ldots & A_{m,n} \end{pmatrix} $ $A_{i,j}$ denotes the entry in row $i$, column $j$. The $k^{th}$ column of $A$ is the $m\text{-by-}1$ matrix $ \begin{pmatrix} A_{1,k} \\ \vdots{} \\ A_{m,k} \end{pmatrix} $ $A_{\cdot,k}$ denotes the $k^{th}$ column of $A$. And the $k^{th}$ row of $A$ is the $1\text{-by-}n$ matrix $ \begin{pmatrix} A_{k,1} & \ldots & A_{k,n} \end{pmatrix} $ $A_{k,\cdot}$ denotes the $k^{th}$ row of $A$.