Matrix/IxJ/nxm/Definition

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}I}$ and ${\displaystyle {}J}$ denote index sets. An ${\displaystyle {}I\times J}$-matrix is a mapping

${\displaystyle I\times J\longrightarrow K,(i,j)\longmapsto a_{ij}.}$

If ${\displaystyle {}I=\{1,\ldots ,m\}}$ and ${\displaystyle {}J=\{1,\ldots ,n\}}$, then we talk about an ${\displaystyle {}m\times n}$-matrix. In this case, the matrix is usually written as

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}.}$