Matrix

A matrix $A$ of size $n \times m$ over a ring $\mathbb{F}$ (for simplicity think $\mathbb{R}$), for $n, m \in \mathbb{Z}_{>0}$ is a collection of $nm$ elements $a_{i,j} \in \mathbb{F}$ with $1 \leq i \leq n$ and $1 \leq j \leq m$. This is usually represented as a rectangle of numbers

$$ > A = \left ( \begin{array} > a_{1,1} & a_{1,2} & \cdots & a_{1,m}\\ > a_{2,1} & a_{2,2} & \cdots & a_{2,m}\\ > \vdots & \vdots & \ddots & \vdots\\ > a_{n,1} & a_{n,2} & \cdots & a_{n,m} > \end{array} \right ) . > $$

Such matrices will be referred to as $M_{n \times m}(\mathbb{F})$. They come with canonical operations such as matrix addition and matrix multiplication (when $n = m$).