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Annihilator Theorem-1

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.

We have

W⁰={f∊V’ :f(α)=0 ∀ α∊W}………..(1)

And W⁰⁰={α∊V : f(α)=0 ∀ f∊W⁰}……..(2)

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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}

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Question:-Let W₁ and W₂ be subspaces of a finite dimensional vector space V.

(a)Prove that (W₁+W₂)^o=W₁⁰⋂W₂⁰

(b)Prove that (W₁⋂W₂)⁰=W₁⁰+W₂⁰.

Solution:-

(a) First we shall prove that

W₁⋂W₂⊆(W₁+W₂)⁰.