Let $k$ be a field and let $m,n\in\mathbb{N}$. For each pair $(i,j)$ with $1\le i\le m$ and $1\le j\le n$, let $E_{ij}\in M_{m\times n}(k)$ denote the matrix whose $(r,s)$-entry is $1_k$ if $(r,s)=(i,j)$ and $0_k$ otherwise. Then the family $\{E_{ij}:1\le i\le m,\ 1\le j\le n\}$ is a basis of the $k$-[vector space](/page/Vector%20Space) $M_{m\times n}(k)$. Consequently, $\dim_k M_{m\times n}(k)=mn$.