
# 6.4: Homomorphisms

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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

1. Monomorphism iff $$T$$ is injective;
2. Epimorphism iff $$T$$ is surjective;
3. Isomorphism iff $$T$$ is bijective;
4. Endomorphism iff $$V=W$$;
5. Automorphism iff $$V=W$$ and $$T$$ is bijective.

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