# linear isomorphism

###### Definition 1.

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

## Properties

1. 1.

Compositions and of linear isomorphisms is a linear isomorphism.

2. 2.

The inverse of a linear isomorphisms is a linear isomorphism.

3. 3.

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

Title linear isomorphism LinearIsomorphism 2013-03-22 14:36:42 2013-03-22 14:36:42 matte (1858) matte (1858) 7 matte (1858) Definition msc 15A04 invertible linear map bijective linear map non-singular linear map