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 S^o 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⟺L_𝛼. Therefore we may regard S⁰⁰ as a subspace of V.Thus

S⁰⁰={α∊V : f(α)=0 ⩝ f∊S⁰}

Theorem:-

Let V be a finite dimensional  vector space over the field F and let W be a subspace over the field F and let W be a subspace of V.Then W⁰⁰=W.

Proof:-

We have

W⁰={f∊V’ :f(𝛼)=0 ⩝ 𝛼⩝W}………..(1)

And             W⁰⁰={𝛼∊V : f(𝛼)=0 ⩝ f∊W⁰}……..(2)

Let 𝛼∊W.Then from(1),f(𝛼)=0 ⩝f∊W⁰ and so from (2), and so from (2), 𝛼∊W⁰⁰

∵                     𝛼∊W ⇨ 𝛼∊W⁰⁰.

Thus W⊆W⁰⁰. Now W is a subspace of W⁰⁰ is also a subspace of V. Since W⊆W⁰⁰,therefore W is a subspace W⁰⁰.

Now dimW+dimW⁰=dimV                      (By theorem)

Applying the same theorem for the vector space V’ and its subspace W⁰,we get

dimW⁰+dim W⁰⁰=dim V’=dim V.

dim W=dim V-dim W⁰ =dim V-[dim V-dim W⁰⁰]

=dim W⁰⁰.

Since W is asubspace of W⁰⁰ and dim W=dim W⁰⁰,therefore

W=W⁰⁰