Math 2.0

Linear and multilinear algebra; matrix theory Definition

Linear map of a matrix

Builds on Matrix of a linear map

Definition

Let FF be a field, and let VV and WW be vector spaces over FF with bases B=(b1,…,bn)B = (b_1, \dots, b_n) and C=(c1,…,cm)C = (c_1, \dots, c_m). Let A=(aij)A = (a_{ij}) be an m×nm \times n matrix over FF. The linear map of AA in the bases BB and CC is the linear map TA ⁣:V→WT_A \colon V \to W with TA(bj)=∑i=1maijci(1≤j≤n).T_A(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).

Remarks

The map A↦TAA \mapsto T_A is the inverse of the map T↦[T]C,BT \mapsto [T]_{C,B} of the matrix of a linear map. For V=FnV = F^n and W=FmW = F^m with the standard bases, TAT_A is the map x↦Axx \mapsto Ax, see multiplication by a matrix as a linear map. In Mathlib, this map is Matrix.toLin.

Reading path

To understand this page from the ground up, read these 11 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 vectordefinitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionLinear map of a matrix, definition, no evidence levelLinear map of a matrixdefinition
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. Matrix of a linear map definition
  11. Linear map of a matrix definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Transformation_matrix, lead section
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