I see the terms linear transformation and (vector space) homomorphism used more or less interchangeably, and the set (space) of linear transformations from V to W referred to as Hom(V, W) or equivalently as L(V, W).

From category theory, a homomorphism is a mapping between objects that preserves structure, so a vector space homomorphism is a linear transformation as this is necessary to preserve the structure of vector addition and scalar multiplication.

But to say a linear transformation is a homomorphism seems to be an invalid assumption. For example, with real / complex inner product spaces only orthogonal / unitary linear transformations preserve the inner product structure and therefore deserve to be considered as homomorphisms.

So, should linear transformations always be referred to as such (not homomorphisms), and the set of linear transformations from V to W as L(V, W), not Hom(V, W) ?

What then about a vector space isomorphism ?

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    $\begingroup$ Morphism of the category of vector spaces is linear transformation. Of course, you can restrict your attention to maps preserving an inner product, but it gives different category. $\endgroup$ – Hanul Jeon May 9 '15 at 11:43
  • $\begingroup$ Indeed, the word "morphism" is highly context-dependent. So the category of vector spaces is not the same as a category of inner product spaces. A morphism of inner-product spaces is a linear transformation that preserves inner products, and is thus a morphism in the category of vector spaces, but not the other way around. $\endgroup$ – pjs36 May 9 '15 at 11:46
  • $\begingroup$ I thought that in category theory a homomorphism is just a morphism. On the other hand, in group theory, or when considering model theoretic notions, a homomorphism is a structure preserving function. $\endgroup$ – Asaf Karagila May 9 '15 at 11:46
  • $\begingroup$ @AsafKaragila Thanks for feedback. I understand that there is some "generic" definition of homomorphism which then devolves down to groups, modules, vector spaces, etc. If this doesn't arise from category theory, is it correct that it arises from "abstract algebra" ? $\endgroup$ – Tom Collinge May 9 '15 at 21:12
  • $\begingroup$ Yes, I think. It'd think it arises from model theory, but many people don't think about it this way, and homomorphisms are considered algebraic-related functions. But don't ask me, I'm a set theorist. :-) $\endgroup$ – Asaf Karagila May 9 '15 at 21:40

They are exactly homomorphisms of vector spaces (preserving linear structures). $\mathbb{R}^n$ and $\mathbb{C}^n$ with inner products are not plain vector spaces, they are Euclidean/Hilbert spaces. So their homomorphism would need to preserve inner products also (hence orthogonal/unitary).

Regarding isomorphisms, there is a theorem that $f:V\to W$ is an isomorphism if and only if $\dim V=\dim W$. And in that view, we often study isomorphisms of the form $f:V\to V$. These are denoted by $\mathrm{End}(V)$, the endomorphisms of $V$.

  • $\begingroup$ You mean there exists an isomorphisms iff they have the same dimension. Not all the homomorphisms need to be isomorphisms. $\endgroup$ – Tobias Kildetoft May 9 '15 at 12:43

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