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Mathematics LibreTexts

6.4 Homomorphisms

  • Page ID
    274
  • [ "article:topic", "vettag:vet4", "targettag:lower", "authortag:schilling", "authorname:schilling" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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.