linear isomorphism

Definition 1.

Suppose $V$ and $W$ are vector spaces and $L\mathrm{:}V\mathrm{\to }W$ is a linear map. Then $L$ is a linear isomorphism if $L$ is bijective.

Properties

1. Compositions and of linear isomorphisms is a linear isomorphism.

2. The inverse of a linear isomorphisms is a linear isomorphism.

3. If either $V$ or $W$ if finite dimensional, then $dimV=dimW$.(This is a consequence of the rank-nullity theorem.)