Math 2.0

Linear and multilinear algebra; matrix theory Definition

Linear isomorphism

Builds on Linear map

Definition

Let FF be a field and let VV and WW be vector spaces over FF. A linear isomorphism from VV to WW is a bijective linear map T ⁣:V→WT \colon V \to W. The spaces VV and WW are isomorphic, written V≅WV \cong W, when a linear isomorphism V→WV \to W exists.

Remarks

The inverse map T−1 ⁣:W→VT^{-1} \colon W \to V of a linear isomorphism is linear. A linear map is an isomorphism if and only if it sends some basis, or each basis, of VV to a basis of WW. For an n×nn \times n matrix AA over FF, the map x↦Axx \mapsto Ax is an isomorphism of FnF^n if and only if AA is invertible. Finite-dimensional spaces are isomorphic if and only if they have equal dimension.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear isomorphism, definition, no evidence levelLinear isomorphismdefinition
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  1. Vector space definition
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Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_map, Definition and first consequences
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