**Theorem:** The set \(M_{m imes n}(\mathbb{F})\) of all \(m imes n\) 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:** \(\dim(M_{m imes n}) = mn\).
**Standard basis:** The matrices \(E_{ij}\) with a 1 in position \((i,j)\) and 0 elsewhere. There are \(mn\) such matrices.
**Example:** \(M_{2 imes 2}(\mathbb{R})\) has dimension 4 with basis:
\[egin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\]