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Linear and multilinear algebra; matrix theory Definition

Determinant of a family of vectors in a basis

Builds on DeterminantBasisCoordinate vector

Definition

Let FF be a field, let VV be a vector space over FF, and let B=(b1,…,bn)B = (b_1, \dots, b_n) be a basis of VV. For vectors v1,…,vn∈Vv_1, \dots, v_n \in V, let MM be the n×nn \times n matrix whose column jj is the coordinate vector [vj]B[v_j]_B. The determinant of (v1,…,vn)(v_1, \dots, v_n) in the basis BB is det⁡B(v1,…,vn)=det⁡M.\det_B(v_1, \dots, v_n) = \det M.

Remarks

The map det⁡B ⁣:Vn→F\det_B \colon V^n \to F is the unique alternating multilinear form with det⁡B(b1,…,bn)=1\det_B(b_1, \dots, b_n) = 1. The vectors v1,…,vnv_1, \dots, v_n form a basis of VV if and only if det⁡B(v1,…,vn)≠0\det_B(v_1, \dots, v_n) \ne 0. For another basis B′B', det⁡B′=det⁡B′(b1,…,bn)⋅det⁡B\det_{B'} = \det_{B'}(b_1, \dots, b_n) \cdot \det_B.

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  1. Identity matrix object
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  4. Square matrix definition
  5. Linear combination definition
  6. Linear map definition
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  8. Linear independence definition
  9. Linear span definition
  10. Multilinear map definition
  11. Spanning set definition
  12. Alternating multilinear map definition
  13. Basis definition
  14. Determinant definition
  15. Coordinate vector definition
  16. Determinant of a family of vectors in a basis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Determinant of n vectors
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Kinddefinition
WritingW0 stub
MSC15A15, 15A75
DefinesDeterminant of a family of vectors in a basis
UsesDeterminant, Basis, Coordinate vector

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