Math 2.0

statement theorem

Matrix of a composition is the product of matrices

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}.

Explanations

L1 human

Doing one linear map after another corresponds to multiplying their matrices. The order is the same as in T∘ST \circ S: the matrix of the map that acts first is on the right. This is why matrix multiplication has its unusual row-times-column rule.

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To read this node, read these 14 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 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 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. 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

Sources

  1. Web https://en.wikipedia.org/wiki/Transformation_matrix, Composing and inverting transformations

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