Space 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 $n(n+1)/2$.

Explanations

L1 human

Matrices of one fixed size can be added and scaled, so they form a vector space. A $2 \times 2$ matrix holds four numbers, so this space has dimension $4$. It is $F^4$ written as a square instead of a row.