PlanetMath: linear isomorphism

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linear isomorphism

(Definition)

Definition 1 Suppose and are vector spaces and $L\colon U\to V$ is a linear map. Then is a linear isomorphism if is bijective.

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

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"linear isomorphism" is owned by matte.

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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

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Cross-references: rank-nullity theorem, consequence, finite dimensional, inverse, compositions, bijective, linear map, vector spaces
There are 17 references to this entry.

This is version 3 of linear isomorphism, born on 2004-09-17, modified 2004-10-17.
Object id is 6186, canonical name is LinearIsomorphism.
Accessed 6082 times total.

Classification:

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

Pending Errata and Addenda

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