Exercise 1216Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

The space of linear maps

#linear maps#vector space structure#dimension

Statement

Let $E$ and $F$ be finite-dimensional vector spaces, with $\dim E=n$ and $\dim F=m$ . Show that $\mathcal{L}(E,F)$ is a vector space of dimension $mn$ .

Hint

Fix bases of $E$ and $F$ ; the map $f\mapsto$ matrix of $f$ is a linear isomorphism onto $\mathcal{M}_{m,n}(\mathbb{R})$ .

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Solution

Vector space structure. For $f,g\in\mathcal{L}(E,F)$ and $\lambda\in\mathbb{R}$ , define $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x)$ . These are linear maps, and the axioms hold pointwise.

Isomorphism. Fix a basis $\mathcal{B}$ of $E$ and a basis $\mathcal{C}$ of $F$ . The map $\Phi:f\mapsto[f]_{\mathcal{B}}^{\mathcal{C}}$ (matrix of $f$ in these bases) is linear and bijective, with inverse $M\mapsto$ the linear map with matrix $M$ .

Dimension. $\dim\mathcal{M}_{m,n}(\mathbb{R})=mn$ , so $\dim\mathcal{L}(E,F)=mn$ .

Conclusion. $\mathcal{L}(E,F)$ is a vector space of dimension $mn$ .

The space of linear maps

Statement

Solution

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