Math 2.0

Linear and multilinear algebra; matrix theory Definition

Matrix of a linear map

Builds on Linear mapBasisCoordinate vector

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 T ⁣:V→WT \colon V \to W be a linear map. The matrix of TT in the bases BB and CC is the m×nm \times n matrix [T]C,B=(aij)[T]_{C,B} = (a_{ij}) defined by T(bj)=∑i=1maijci(1≤j≤n).T(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n). Equivalently, column jj of [T]C,B[T]_{C,B} is the coordinate vector [T(bj)]C[T(b_j)]_C.

Remarks

For each v∈Vv \in V, [T(v)]C=[T]C,B[v]B[T(v)]_C = [T]_{C,B} [v]_B. The map T↦[T]C,BT \mapsto [T]_{C,B} is a linear isomorphism Hom⁡F(V,W)→Mm×n(F)\Hom_F(V, W) \to M_{m \times n}(F). For the rotation of ℝ2\R^2 by an angle θ\theta, in the standard basis, the matrix is (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}. Mathlib writes this map as LinearMap.toMatrix.

Reading path

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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 mapdefinition
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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

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Transformation_matrix, lead section
About this page
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Kinddefinition
WritingW0 stub
MSC15A04
DefinesMatrix of a linear map
UsesLinear map, Basis, Coordinate vector

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