Math 2.0

Linear and multilinear algebra; matrix theory Definition

Matrix of a bilinear form

Builds on Bilinear formBasis

Definition

Let FF be a field and let VV be a vector space over FF with a basis (e1,…,en)(e_1, \dots, e_n). Let BB be a bilinear form on VV. The matrix of BB in this basis is the n×nn \times n matrix MM with entries Mij=B(ei,ej)(1≤i,j≤n).M_{ij} = B(e_i, e_j) \qquad (1 \le i, j \le n).

Remarks

If xx and yy are the coordinate vectors of uu and vv, then B(u,v)=xTMyB(u, v) = x^T M y. The map B↦MB \mapsto M is a linear bijection from the bilinear forms on VV to the n×nn \times n matrices over FF. For the dot product on ℝn\R^n and the standard basis, MM is the identity matrix.

Reading path

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Vector space, definition, no evidence levelVector spacedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear 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 levelBasisdefinitionMatrix of a bilinear form, definition, no evidence levelMatrix of a bilinear …definition
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. Bilinear form definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Matrix of a bilinear form definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bilinear_form, Coordinate representation
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