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annihilator of vector subspace

(Definition)

If is a vector space, and is any subset of , the annihilator of , denoted by , is the subspace of the dual space that kills every vector in :

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

Similarly, if $\Lambda$ is any subset of , the annihilated subspace of $\Lambda$ is

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

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

Properties

Assume

and $\Phi$ denote subspaces of

and

, respectively, and let $\widehat{\:}$ denote the natural isomorphism from

to its double dual $V^{**}$ .

i.: $S^0 = \left({\mathrm{span}}S\right)^0$
ii.: $\Lambda^{-0} = \left({\mathrm{span}}\Lambda \right)^{-0}$
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 denotes the sum of two subspaces of .
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 .