# 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 $S$:

 $S^{0}=\{\phi\in V^{*}:\phi(v)=0\textrm{ for all }v\in S\}\,.$

Similarly, if $\Lambda$ is any subset of $V^{*}$, the annihilated subspace of $\Lambda$ is

 $\Lambda^{-0}=\{v\in V:\phi(v)=0\textrm{ for all }\phi\in\Lambda\}=\bigcap_{% \phi\in\Lambda}\ker\phi\,.$

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

## 1 Properties

Assume $V$ is finite-dimensional. Let $W$ and $\Phi$ denote subspaces of $V$ and $V^{*}$, respectively, and let $\widehat{\>}$ denote the natural isomorphism from $V$ to its double dual $V^{**}$.

1. i.

$S^{0}=\left(\operatorname{span}S\right)^{0}$

2. ii.

$\Lambda^{-0}=\left(\operatorname{span}\Lambda\right)^{-0}$

3. iii.

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

4. iv.

$\left(\Phi^{-0}\right)^{0}=\Phi$

5. v.

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

6. vi.

$\dim W+\dim W^{0}=\dim V$ (a dimension theorem)

7. vii.

$\dim\Phi+\dim\Phi^{-0}=\dim V^{*}=\dim V$

8. 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 $V$.

9. ix.

If $T:V\to V$ is a linear operator, and $W=\ker T$, then the image of the pullback $T^{*}:V^{*}\to V^{*}$ is $W^{0}$.

## References

Title annihilator of vector subspace AnnihilatorOfVectorSubspace 2013-03-22 15:25:59 2013-03-22 15:25:59 stevecheng (10074) stevecheng (10074) 5 stevecheng (10074) Definition msc 15A03 annihilator annihilated subspace