Math 2.0

statement proposition

Trace of a linear map is the trace of its matrix

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).

Explanations

L1 human

We can compute the trace of a linear map in any basis that we like. We write the matrix in that basis and add its diagonal entries. Every basis gives the same number.

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What this needs

To read this node, read these 16 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  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

Sources

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace of a linear operator

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