Annihilators:-
If V is a
vector space over the field F and S is a subset of V,the annihilator of S is
the set So of all linear functional f on V such that
f(α)=0 ∀ α∊S
Sometimes A(S) is also used to denote the
annihilators of S.
Thus S0={f∊V’: f(α)=0 ∀ α∊S}
Annihilator of an anhilator:-
Let V be a vector space over the field F.If
S is any subset of V,then S0 is a subspace of V’.By definition of an
annihilator, we have
(S0)0=S00={L∊V’:
L(f)=0 ∀ f∊S0}
Obviously S00 is a subspace of
V’’.But if V is finite dimensional then we have identified V’’ with V through
the natural isomorphism α⇔Lα. Therefore we may
regard S00 as a subspace of V.Thus
S00={α∊V
: f(α)=0 ∀ f∊S0}
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