The set of m by n matrices is a vector space.

**Theorem:** The set \(M_{m \times n}(\mathbb{F})\) of all \(m \times n\) matrices with entries in \(\mathbb{F}\) is a vector space over \(\mathbb{F}\).

- **Addition:** \((A + B)_{ij} = A_{ij} + B_{ij}\) (entrywise)
- **Scalar multiplication:** \((cA)_{ij} = c \cdot A_{ij}\)

**Zero vector:** The zero matrix (all entries are 0).

**Dimension:** \(\dim(M_{m \times n}) = mn\).

**Standard basis:** The matrices \(E_{ij}\) with a 1 in position \((i,j)\) and 0 elsewhere. There are \(mn\) such matrices.

**Example:** \(M_{2 \times 2}(\mathbb{R})\) has dimension 4 with basis:
\[\begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\]Copy