Math 2.0

Linear and multilinear algebra; matrix theory Definition

Dual basis

Builds on Dual spaceBasis

Definition

Let FF be a field and let VV be a finite-dimensional vector space over FF. Let B=(b1,…,bn)B = (b_1, \dots, b_n) be a basis of VV. The dual basis of BB is the family (b1∗,…,bn∗)(b_1^*, \dots, b_n^*) of elements of the dual space V∗V^*. These elements are defined by bi∗(bj)=δij(1≤i,j≤n),b_i^*(b_j) = \delta_{ij} \qquad (1 \le i, j \le n), where δij=1\delta_{ij} = 1 if i=ji = j and δij=0\delta_{ij} = 0 if i≠ji \ne j.

Remarks

The dual basis is a basis of V∗V^*. The form bi∗b_i^* returns coordinate ii: bi∗(a1b1+⋯+anbn)=aib_i^*(a_1 b_1 + \cdots + a_n b_n) = a_i. Each f∈V∗f \in V^* is f=∑if(bi) bi∗f = \sum_i f(b_i)\, b_i^*. For the standard basis of FnF^n, ei∗e_i^* is the map x↦xix \mapsto x_i.

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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 form, definition, no evidence levelLinear formdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionDual basis, definition, no evidence levelDual basisdefinition
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  1. Vector space definition
  2. Linear combination definition
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  6. Linear form definition
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  8. Space of linear maps definition
  9. Dual space definition
  10. Spanning set definition
  11. Basis definition
  12. Dual basis definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_basis, Finite-dimensional vector spaces
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DefinesDual basis
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