annihilator of vector subspace

If V is a vector spaceMathworldPlanetmath, and S is any subset of V, the annihilatorPlanetmathPlanetmathPlanetmathPlanetmath of S, denoted by S0, is the subspacePlanetmathPlanetmath of the dual spaceMathworldPlanetmathPlanetmath V* that kills every vector in S:

S0={ϕV*:ϕ(v)=0 for all vS}.

Similarly, if Λ is any subset of V*, the annihilated subspace of Λ is

Λ-0={vV:ϕ(v)=0 for all ϕΛ}=ϕΛkerϕ.

(Note: this may not be the standard notation.)

1 Properties

Assume V is finite-dimensional. Let W and Φ denote subspaces of V and V*, respectively, and let ^ denote the natural isomorphism from V to its double dual V**.

  1. i.


  2. ii.


  3. iii.


  4. iv.


  5. v.


  6. vi.

    dimW+dimW0=dimV (a dimension theorem)

  7. vii.


  8. viii.

    (W1+W2)0=W10W20, where W1+W2 denotes the sum of two subspaces of V.

  9. ix.

    If T:VV is a linear operator, and W=kerT, then the image of the pullback T*:V*V* is W0.


Title annihilator of vector subspace
Canonical name AnnihilatorOfVectorSubspace
Date of creation 2013-03-22 15:25:59
Last modified on 2013-03-22 15:25:59
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 5
Author stevecheng (10074)
Entry type Definition
Classification msc 15A03
Defines annihilator
Defines annihilated subspace