Math 2.0

statement proposition

Matrix of the dual map is the transpose

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.

Explanations

L1 human

The dual map is the linear algebra form of the transpose of a matrix. In dual bases, its matrix is the transpose of the matrix of the original map. This explains why rows and columns play symmetric roles.

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To read this node, read these 18 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 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 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.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Transpose_of_a_linear_map, Representation as a matrix

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