Math 2.0

definition

Annihilator of a 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.

Explanations

L1 human

The annihilator of a subspace collects the linear forms that are zero on all of it. In R3\R^3, the forms that vanish on a plane through the origin are the multiples of one form, whose kernel is that plane. A larger subspace has a smaller annihilator.

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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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Sources

  1. Web https://en.wikipedia.org/wiki/Dual_space, Quotient spaces and annihilators

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