PlanetMath: linear isomorphism

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

linear isomorphism

(Definition)

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.

Compositions and of linear isomorphisms is a linear isomorphism.
The inverse of a linear isomorphisms is a linear isomorphism.
If either $V$ or $W$ if finite dimensional, then $\dim V=\dim W$ . (This is a consequence of the rank-nullity theorem.)

Anyone with an account can edit this entry. Please help improve it!

"linear isomorphism" is owned by matte. [ full author list (2) ]

(view preamble | get metadata)

Other names:

invertible linear map, bijective linear map, non-singular linear map

This object's parent.

Attachments:: I-AB is invertible if and only if I-BA is invertible (Theorem) by asteroid

Log in to rate this entry.
(view current ratings)

Cross-references: rank-nullity theorem, consequence, finite dimensional, inverse, compositions, bijective, linear map, vector spaces
There are 23 references to this entry.

This is version 4 of linear isomorphism, born on 2004-09-17, modified 2009-03-30.
Object id is 6186, canonical name is LinearIsomorphism.
Accessed 9010 times total.

Classification:

15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact