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Mathematics LibreTexts

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

  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.