annihilator of vector subspace
If $V$ is a vector space^{}, and $S$ is any subset of $V$, the annihilator^{} of $S$, denoted by ${S}^{0}$, is the subspace^{} of the dual space^{} ${V}^{*}$ that kills every vector in $S$:
$${S}^{0}=\{\varphi \in {V}^{*}:\varphi (v)=0\text{for all}v\in S\}.$$ 
Similarly, if $\mathrm{\Lambda}$ is any subset of ${V}^{*}$, the annihilated subspace of $\mathrm{\Lambda}$ is
$${\mathrm{\Lambda}}^{0}=\{v\in V:\varphi (v)=0\text{for all}\varphi \in \mathrm{\Lambda}\}=\bigcap _{\varphi \in \mathrm{\Lambda}}\mathrm{ker}\varphi .$$ 
(Note: this may not be the standard notation.)
1 Properties
Assume $V$ is finitedimensional. Let $W$ and $\mathrm{\Phi}$ denote subspaces of $V$ and ${V}^{*}$, respectively, and let $\widehat{}$ denote the natural isomorphism from $V$ to its double dual ${V}^{**}$.

i.
${S}^{0}={\left(\mathrm{span}S\right)}^{0}$

ii.
${\mathrm{\Lambda}}^{0}={\left(\mathrm{span}\mathrm{\Lambda}\right)}^{0}$

iii.
${W}^{00}=\widehat{W}$

iv.
${\left({\mathrm{\Phi}}^{0}\right)}^{0}=\mathrm{\Phi}$

v.
${\left({W}^{0}\right)}^{0}=W$

vi.
$dimW+dim{W}^{0}=dimV$ (a dimension theorem)

vii.
$dim\mathrm{\Phi}+dim{\mathrm{\Phi}}^{0}=dim{V}^{*}=dimV$

viii.
${({W}_{1}+{W}_{2})}^{0}={W}_{1}^{0}\cap {W}_{2}^{0}$, where ${W}_{1}+{W}_{2}$ denotes the sum of two subspaces of $V$.

ix.
If $T:V\to V$ is a linear operator, and $W=\mathrm{ker}T$, then the image of the pullback ${T}^{*}:{V}^{*}\to {V}^{*}$ is ${W}^{0}$.
References
 1 Friedberg, Insel, Spence. Linear Algebra. PrenticeHall, 1997.
Title  annihilator of vector subspace 

Canonical name  AnnihilatorOfVectorSubspace 
Date of creation  20130322 15:25:59 
Last modified on  20130322 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 