Matrix/IxJ/nxm/Definition

Matrix

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

$I\times J⟶K,(i,j)⟼a_{ij}.$

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

$\left(\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right).$