Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Trace of a linear map is the trace of its matrix

Builds on Trace 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 trace of TT equals the trace of the matrix of TT in BB: tr⁡T=tr⁡[T]B.\tr T = \tr [T]_B.

Remarks

The key fact is that similar matrices have the same trace, by tr⁡(AB)=tr⁡(BA)\tr(AB) = \tr(BA). With a basis-free definition, the trace is the map End⁡F(V)≅V∗⊗V→F\End_F(V) \cong V^* \otimes V \to F induced by evaluation f⊗v↦f(v)f \otimes v \mapsto f(v).

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Matrix, 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 subspacedefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionTrace of a linear map, definition, no evidence levelTrace of a linear mapdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionTrace of a linear map is the trace of its matrix, statement, no evidence levelTrace of a linear map…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.
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  1. Matrix definition
  2. Vector space definition
  3. Square matrix definition
  4. Linear combination definition
  5. Linear map definition
  6. Linear subspace definition
  7. Trace of a matrix definition
  8. Linear independence definition
  9. Endomorphism definition
  10. Linear span definition
  11. Trace of a linear map definition
  12. Spanning set definition
  13. Basis definition
  14. Coordinate vector definition
  15. Matrix of a linear map definition
  16. Trace of a linear map is the trace of its matrix statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace of a linear operator
About this page
Tag0WCPH5Z7FA
URIm2://0WCPH5Z7FA
Kindstatement, proposition
WritingW0 stub
MSC15A04, 15A15
StatusThis copy of the site has no build data. Run m2 status 0WCPH5Z7FA.
UsesTrace of a linear map, Matrix of a linear map

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