Linear and multilinear algebra; matrix theory Example

Space of matrices

Builds on Matrix Matrix addition Scalar multiplication of matrices

Example

Let $F$ be a field and let $m, n \ge 1$ . The set $M_{m \times n}(F)$ of $m \times n$ matrices over $F$ is a vector space over $F$ , with matrix addition and scalar multiplication. Its zero vector is the zero matrix. The matrices $E_{ij}$ , with entry $1$ in position $(i, j)$ and $0$ elsewhere, form a basis, so $\dim M_{m \times n}(F) = mn$ .

Remarks

The map that lists the entries of a matrix row by row is a linear isomorphism $M_{m \times n}(F) \to F^{mn}$ . The space $M_{m \times n}(F)$ is isomorphic to the space of linear maps from $F^n$ to $F^m$ . The symmetric $n \times n$ matrices form a subspace of dimension $n(n+1)/2$ .

Reading path

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Matrix definition
Matrix addition definition
Scalar multiplication of matrices definition
Space of matrices example

Tag	`0RE6X3BGPN`
URI	`m2://0RE6X3BGPN`
Kind	example
Writing	W0 stub
MSC	15A03
Instance of	Vector space
Uses	Matrix, Matrix addition, Scalar multiplication of matrices

Space of matrices

Example

Remarks

Reading path

Further reading

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