# 6.4 Homomorphisms

It should be mentioned that linear maps between vector spaces are also called **vector space ****homomorphisms**. Instead of the notation \( \mathcal{L} (V,W) \), one often sees the convention

\[ \mathrm{Hom}_\mathbb{F} (V,W) = \{ T:V \to W \mid \text{ T is linear} \}. \]

A homomorphism \(T:V \to W \) is also often called

**Monomorphism**iff \(T \) is injective;**Epimorphism**iff \(T \) is surjective;**Isomorphism**iff \(T \) is bijective;**Endomorphism**iff \(V=W \);**Automorphism**iff \(V=W \) and \(T \) is bijective.

### Contributors

- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis

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