# What is a bijective linear mapping called?

Friedberg - Linear Algebra p.102

This book states that "a bijective linear map from a vector space to another vector space is called an isomorphism".

As far as know, generally isomorphism means bijective homomorphism and notion for this is $\cong$, NOT bijective linear map.

What is bijective linear map called? And what is the notion for this?

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I prefer the category theoretic definition of isomorphism: An isomorhism is a morphism with a (left and right) invers morphism. Note for example that a merely bijective continuous map is not considered a topologicyl isomorphism (=homeomorphism). Alternatively, one always has to keep all those induced maps on the structure in mind ... – Hagen von Eitzen Mar 2 '13 at 12:20
You might prefer it, but I think that Katlus is not into morphisms yet. – user62182 Mar 2 '13 at 12:28

The concepts are equivalent. You can show that in thi situation, a linear operator is a homomorphism, therefore, a bijective linear operator is a isomorphism.

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@user62183 Thank you. It's off the topic, but how do i write "isomorphic into" and "isomorphic onto" in latex? Which look like --< and >--. – Jj- Mar 2 '13 at 12:14
"In this situation" meaning "in the category of vector spaces" – Hagen von Eitzen Mar 2 '13 at 12:16
@Katlus try searching here: google.com/… – user62182 Mar 2 '13 at 12:28

More generally speaking, an isomorphism is a bijection between two objects that preserves their structure. The meaning of structure depends on the category you're working in.

• Sets: An isomorphism of sets is just a bijective map.
• Topological spaces: An isomorphism is a bijective continuous map whose inverse is continuous (thus, preserves open sets).
• Groups: An isomorphism is a bijective map that's a homomorphism (thus, preserves the group operation).
• Vector spaces: An isomorphism is a bijective map that's a linear transformation (thus, preserves the linear structure).

The list goes on. In your case, I will add that many times a vector space also has a topology (such is the case with $\mathbb R^n$, for example). In this case you'd be interested in an isomorphism of topological vector spaces, that is, a bijective map that's linear, continuous, with continuous inverse. It turns out that some of these requirements are superfluous :)

As for the notation for isomorphic spaces - I don't know if there's a standard one. It seems every book has its own favorite version of the equality symbol.

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