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

Put content here.**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 \(\mathbb{F}\).
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- **Addition:** \((A + B)_{ij} = A_{ij} + B_{ij}\) (entrywise)
- **Scalar multiplication:** \((cA)_{ij} = c \cdot A_{ij}\)
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**Zero vector:** The zero matrix (all entries are 0).
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**Dimension:** \(\dim(M_{m 	imes n}) = mn\).
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**Standard basis:** The matrices \(E_{ij}\) with a 1 in position \((i,j)\) and 0 elsewhere. There are \(mn\) such matrices.
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**Example:** \(M_{2 	imes 2}(\mathbb{R})\) has dimension 4 with basis:
\[egin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\]

# Parents

* Examples of vector spaces