annihilator of vector subspace
If V is a vector space, and S is any subset of V,
the annihilator
of S, denoted by S0,
is the subspace
of the dual space
V*
that kills every vector in S:
S0={ϕ∈V*:ϕ(v)=0 for all v∈S}. |
Similarly, if Λ is any subset of V*, the annihilated subspace of Λ is
Λ-0={v∈V:ϕ(v)=0 for all ϕ∈Λ}=⋂ϕ∈Λkerϕ. |
(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**.
-
i.
S0=(spanS)0
-
ii.
Λ-0=(spanΛ)-0
-
iii.
W00=ˆW
-
iv.
(Φ-0)0=Φ
-
v.
(W0)-0=W
-
vi.
(a dimension theorem)
-
vii.
-
viii.
, where denotes the sum of two subspaces of .
-
ix.
If is a linear operator, and , then the image of the pullback is .
References
- 1 Friedberg, Insel, Spence. Linear Algebra. Prentice-Hall, 1997.
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 |