Math 2.0

Linear and multilinear algebra; matrix theory Definition

Determinant of a linear map

Builds on DeterminantEndomorphismMatrix of a linear map

Definition

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let T ⁣:V→VT \colon V \to V be an endomorphism. The determinant of TT is det⁡T=det⁡[T]B,\det T = \det [T]_B, the determinant of the matrix of TT in a basis BB of VV.

Remarks

The value does not depend on BB, because another basis gives a similar matrix P−1[T]BPP^{-1}[T]_B P, with the same determinant. It satisfies det⁡(S∘T)=det⁡S⋅det⁡T\det(S \circ T) = \det S \cdot \det T and det⁡id⁡V=1\det \id_V = 1. The map TT is invertible if and only if det⁡T≠0\det T \ne 0. For V=0V = 0, the convention is det⁡T=1\det T = 1.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionBasis, definition, no evidence levelBasisdefinitionDeterminant, definition, no evidence levelDeterminantdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionDeterminant of a linear map, definition, no evidence levelDeterminant of a line…definition
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  16. Coordinate vector definition
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  18. Determinant of a linear map definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Determinant of an endomorphism
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Kinddefinition
WritingW0 stub
MSC15A04, 15A15
DefinesDeterminant of a linear map
UsesDeterminant, Endomorphism, Matrix of a linear map

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