Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant of a linear map is the determinant of its matrix

Builds on Determinant of a linear mapMatrix of a linear map

Statement

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

Remarks

The key fact is det⁡(P−1AP)=det⁡A\det(P^{-1} A P) = \det A, which follows from the product rule. Over a commutative ring, the same statement holds for free modules of finite rank.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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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…definitionDeterminant of a linear map is the determinant of its matrix, statement, no evidence levelDeterminant of a line…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear combination definition
  6. Linear map definition
  7. Linear subspace definition
  8. Linear independence definition
  9. Endomorphism definition
  10. Linear span definition
  11. Multilinear map definition
  12. Spanning set definition
  13. Alternating multilinear map definition
  14. Basis definition
  15. Determinant definition
  16. Coordinate vector definition
  17. Matrix of a linear map definition
  18. Determinant of a linear map definition
  19. Determinant of a linear map is the determinant of its matrix statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Determinant of an endomorphism
About this page
Tag0KMN32XGPF
URIm2://0KMN32XGPF
Kindstatement, proposition
WritingW0 stub
MSC15A04, 15A15
StatusThis copy of the site has no build data. Run m2 status 0KMN32XGPF.
UsesDeterminant of a linear map, Matrix of a linear map

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