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)

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

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