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.)

Title	linear isomorphism
Canonical name	LinearIsomorphism
Date of creation	2013-03-22 14:36:42
Last modified on	2013-03-22 14:36:42
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Definition
Classification	msc 15A04
Synonym	invertible linear map
Synonym	bijective linear map
Synonym	non-singular linear map