What is a bijective linear mapping called?

Question

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?

Answer 1

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.

Answer 2

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.