Isomorphism is an equivalence relation between vector spaces ↩

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created: 2021-11-14 20:17:47
modified: 2022-01-11 15:56:45

Statement: Vector space isomorphism is an equivalence relation between vector spaces.

Proof

Reflexivity: Just need to show that any space is isomorphic to itself. For this, let's consider the identity map . It is clearly a bijection and it is a linear map, so it is an isomorphism.

Symmetry: that if is isomorphic to , then also is isomorphic to . This is true due to the fact that isomorphisms are invertible, which yield another isomorphism from to .

Transitivity: that if and then . Let and be isomorphisms. Consider their composition . Because a composition of bijections is a bijection, we only need to check the compositions linearity. And it is easy to see that is a linear map:

Thus the composition is an isomorphism.
QED