Math 2.0

Linear and multilinear algebra; matrix theory Definition

Annihilator of a subspace

Builds on Dual spaceLinear subspace

Definition

Let FF be a field, let VV be a vector space over FF, and let WW be a subspace of VV. The annihilator of WW is the subset of the dual space W0={f∈V∗:f(w)=0 for all w∈W}.W^0 = \{f \in V^* : f(w) = 0 \text{ for all } w \in W\}.

Remarks

The annihilator W0W^0 is a subspace of V∗V^*. If W1⊆W2W_1 \subseteq W_2, then W20⊆W10W_2^0 \subseteq W_1^0. In finite dimensions, dim⁡W+dim⁡W0=dim⁡V\dim W + \dim W^0 = \dim V, see the dimension of the annihilator. The annihilator of the line spanned by (1,1)(1, 1) in F2F^2 is spanned by the form (x,y)↦x−y(x, y) \mapsto x - y.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear form, definition, no evidence levelLinear formdefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionAnnihilator of a subspace, definition, no evidence levelAnnihilator of a subs…definition
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  1. Vector space definition
  2. Linear map definition
  3. Linear subspace definition
  4. Linear form definition
  5. Space of linear maps definition
  6. Dual space definition
  7. Annihilator of a subspace definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_space, Quotient spaces and annihilators
About this page
Tag0R6M158SQ9
URIm2://0R6M158SQ9
Kinddefinition
WritingW0 stub
MSC15A04
DefinesAnnihilator of a subspace
UsesDual space, Linear subspace

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