Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Change of basis formula

Builds on Change of basis matrixMatrix of a linear map

Statement

Let FF be a field and let VV and WW be finite-dimensional vector spaces over FF. Let B,B′B, B' be bases of VV, and let C,C′C, C' be bases of WW. Let PP be the change of basis matrix from BB to B′B', and QQ the change of basis matrix from CC to C′C'. Then, for each linear map T ⁣:V→WT \colon V \to W, [T]C′,B′=Q−1[T]C,BP,[T]_{C',B'} = Q^{-1} [T]_{C,B} P, where [T][T] denotes the matrix of TT in the given bases. In particular, for an endomorphism TT of VV and C=BC = B, C′=B′C' = B', one gets [T]B′=P−1[T]BP[T]_{B'} = P^{-1} [T]_B P.

Remarks

The formula follows from the matrix of a composition, applied to T=id⁡W∘T∘id⁡VT = \id_W \circ T \circ \id_V. For endomorphisms, the two matrices are similar. For linear maps between two spaces, the two matrices are equivalent.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionChange of basis matrix, definition, no evidence levelChange of basis matrixdefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionChange of basis formula, statement, no evidence levelChange of basis formu…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. Vector space definition
  2. Linear combination definition
  3. Linear map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Coordinate vector definition
  10. Change of basis matrix definition
  11. Matrix of a linear map definition
  12. Change of basis formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Change_of_basis, Endomorphisms
About this page
Tag0MQXVZEBMT
URIm2://0MQXVZEBMT
Kindstatement, theorem
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 0MQXVZEBMT.
UsesChange of basis matrix, Matrix of a linear map

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