Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Matrix of the dual map is the transpose

Builds on Dual mapDual basisMatrix of a linear mapTranspose

Statement

Let FF be a field, and let VV and WW be finite-dimensional vector spaces over FF. Let BB be a basis of VV and CC a basis of WW, with dual bases B∗B^* and C∗C^*. Let T ⁣:V→WT \colon V \to W be a linear map with dual map T∗T^*. Then [T∗]B∗,C∗=([T]C,B)T,[T^*]_{B^*, C^*} = \left([T]_{C, B}\right)^T, where [ ⋅ ][\,\cdot\,] is the matrix of a linear map and the exponent TT is the transpose.

Remarks

The proof computes entry (j,i)(j, i) of [T∗][T^*] as T∗(ci∗)(bj)=ci∗(T(bj))T^*(c_i^*)(b_j) = c_i^*(T(b_j)), which is entry (i,j)(i, j) of [T][T]. With the annihilator, this gives a proof that row rank equals column rank.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionTranspose, definition, no evidence levelTransposedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear form, definition, no evidence levelLinear formdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionDual map, definition, no evidence levelDual mapdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionDual basis, definition, no evidence levelDual basisdefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionMatrix of the dual map is the transpose, statement, no evidence levelMatrix of the dual ma…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. Matrix definition
  2. Vector space definition
  3. Transpose definition
  4. Linear combination definition
  5. Linear map definition
  6. Linear subspace definition
  7. Linear independence definition
  8. Linear form definition
  9. Linear span definition
  10. Space of linear maps definition
  11. Dual space definition
  12. Spanning set definition
  13. Basis definition
  14. Dual map definition
  15. Coordinate vector definition
  16. Dual basis definition
  17. Matrix of a linear map definition
  18. Matrix of the dual map is the transpose statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Transpose_of_a_linear_map, Representation as a matrix
About this page
Tag0ZPB3180AY
URIm2://0ZPB3180AY
Kindstatement, proposition
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 0ZPB3180AY.
UsesDual map, Dual basis, Matrix of a linear map, Transpose

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