Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Matrix of a composition is the product of matrices

Builds on Matrix of a linear mapComposition of linear mapsMatrix multiplication

Statement

Let FF be a field, and let UU, VV, WW be finite-dimensional vector spaces over FF with bases AA, BB, CC. Let S ⁣:U→VS \colon U \to V and T ⁣:V→WT \colon V \to W be linear maps. Then [T∘S]C,A=[T]C,B [S]B,A,[T \circ S]_{C,A} = [T]_{C,B}\, [S]_{B,A}, where [ ⋅ ][\,\cdot\,] denotes the matrix of a linear map and the right side is a matrix product.

Remarks

This identity is the reason for the definition of matrix multiplication. It implies that matrix multiplication is associative, because composition of maps is associative. For SS invertible, it gives [S−1]A,B=[S]B,A−1[S^{-1}]_{A,B} = [S]_{B,A}^{-1}.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionComposition of linear maps, definition, no evidence levelComposition of linear…definitionLinear span, definition, no evidence levelLinear spandefinitionSpanning 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 mapdefinitionMatrix of a composition is the product of matrices, statement, no evidence levelMatrix of a compositi…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. Matrix definition
  2. Vector space definition
  3. Matrix multiplication definition
  4. Linear combination definition
  5. Linear map definition
  6. Linear subspace definition
  7. Linear independence definition
  8. Composition of linear maps definition
  9. Linear span definition
  10. Spanning set definition
  11. Basis definition
  12. Coordinate vector definition
  13. Matrix of a linear map definition
  14. Matrix of a composition is the product of matrices statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Transformation_matrix, Composing and inverting transformations
About this page
Tag05Q1B7MH0M
URIm2://05Q1B7MH0M
Kindstatement, theorem
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 05Q1B7MH0M.
UsesMatrix of a linear map, Composition of linear maps, Matrix multiplication

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