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+W2.
∴ (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.
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