Annihilators-Question

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.

Let       f∊W₁⁰⋂W₂⁰.Then f∊W₁⁰, f∊W₂⁰.

Suppose α  is any vector in W₁+W₂.Then

α=α₁+α₂    where α₁∊W₁, α₂∊W₂

We have

f(α)=f(α₁+α₂)=f(α₁)+f(α₂)

=0+0   [∵α₁∊W₁ & f∊W₁⁰ ⇨ f(α₁)=0 and similarly f(α₂)=0]

=0.

Thus  f(α)=0 ⩝ α∊W₁+W₂

∴               f∊(W₁+W₂)⁰.

∴             W₁⁰⋂W₂⁰⊆(W₁+W₂)⁰.                 ………….(1)

Now we shall prove that

(W₁+W₂)⁰   ⊆W₁⁰⋂W₂⁰.

We have   W₁⊆W₁+W_2.

∴                 (W₁+W₂)⁰⊆W₁⁰                         …………..(2)

Similarly, W₂⊆W₁+W₂.

∴                 (W₁+W₂)⁰⊆W₂⁰                        …………..(3)

From (2) & (3),we have

(W₁+W₂)⁰⊆W₁⁰⋂W₂⁰ ……(4)

From (1) & (4),we have

(W₁+W₂)⁰=W₁⁰⋂W₂⁰

(b)Let us use the result (a) for the vector space V’ in place of the vector space V. Thus replacing W₁ and W₂ byW₂⁰ in (a),we get

(W₁⁰+W₂⁰)=W₁⁰⁰⋂W₂⁰⁰

⇨                           (W₁⁰+W₂⁰)=W₁⋂W₂    [∵W₁⁰⁰=W₁ etc.]

⇨                           (W₁⁰+W₂⁰)⁰⁰=(W₁W₂)⁰

⇨                                 W₁⁰+W₂⁰=(W₁+W₂)⁰.