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.)

Properties

i.

$S^0 = \left(\linspan S\right)^0$

ii.

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

iii.

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

iv.

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

v.

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

vi.

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

vii.

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

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 $W = \ker T$ , then the image of the pullback $T^*: V^* \to V^*$ is $W^0$ .

Bibliography

1

Friedberg, Insel, Spence. Linear Algebra. Prentice-Hall, 1997.