Math 2.0

statement proposition

Determinant of a linear map is the determinant 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 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.

Explanations

L1 human

We can compute the determinant of a linear map in any basis. We write its matrix in that basis, and we take the determinant of the matrix. Every basis gives the same answer.

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What this needs

To read this node, read these 19 nodes in this order. The foundations come first, and the list ends with this node.

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 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. 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

Sources

  1. Web https://en.wikipedia.org/wiki/Determinant, Determinant of an endomorphism

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