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)

Question-Linear Transformation

Question:- Show that the mapping T:V₂(R)→V₃(R) defined by T(a,b)=(a+b, a-b, b) is a linear transformation from V₂(R) into V₃(R).

Find the range, rank, null space and null(T) of T.

Solution:- Given that

T:V₂(R)→V₃(R)

Such that T(a,b)=(a+b,a-b,b)

Let a,b∊R and α=(a₁,b₁) & β=(a₁,b₂)

Then aα+bβ= a(a₁,b₁)+b(a₂,b₂)

 =(aa₁,ab₁)+(ba₂,bb₂)

=(aa₁+ba₂, ab₁+bb₂)

∴  T(aα+bβ)= T(aa+ba, ab+bb)

= (aa₁+ba₂+ab₁+bb₂, aa₁+ba₂-ab₁-bb₂, ab₁+bb₂)

=[a(a₁+b₁)+b(a₂+b₂), a(a₁-b₁)+b(a₂-b₂), ab₁+bb₂]

=a(a₁+b₁, a₁-b₁, b₁)+b(a₂+b₂, a₂-b₂, b₂)

=aT(α)+bT(β)

∴  T:V₂(R)→V₃(R) is a linear transformation.

Tingle Inequality

Statement:- In an inner product space over F then prove that ||α+β||≤||α||+||β||.

Proof:- Since, ||α+β||=√(α+β,α+β)         

⇨                    ||α+β||²=(α,β,α,β)

⇨                     ||α+β||²=(α,α+β)+(β,α+β) {by linearity}

=(α,α)+(α,β)+(β,α)+(β,β)

=||α||²+(α,β)+(β,α)+||β||²

=||α||²+(α,β)+( )+||β||²

{since(α,α)=||α||²&(β,α)=( )

=||α||²+2Re(α,β)+||β||²

since , Re(α,β)≤|(α,β)| {since +Z=2x=2Re(z)}

⇨    ||α||²+2Re(α,β)+||β||²≤||α||²+2|(α,β)|+||β||²

||α+β||²≤||α||²+2|(α,β)|+||β||²

≤||α||²+2||α||||β||+||β||²

(since |(α,β)|≤||α||||β||

⇨                                 ||α+β||²≤(||α||+||β||)²

taking square root

⇨                                  ||α+β||≤||α||+||β||      PROVED.