Question

Date: Name (PRINT): 14. Prove either part (a) part (b) of the following (indicate which one you want graded) or (a) Given that W and V are both subspaces of a vector space U, prove that the intersection of W and V is a subspace of of U. ans,

Answer 1

Sol.

Given that W and V are subspaces of a vector space U.

To prove:- W$\cap$ V is a subspace of U.

We will prove it by using Subspace Test.

(i) Since W and V are subspaces of U therefore 0 $\epsilon$ W and 0 $\epsilon$ V

So 0 $\epsilon$ W $\cap$ V

(ii) Let v , w $\epsilon$ W $\cap$ V  

$\implies$ v,w $\epsilon$ W and v,w $\epsilon$ V

since W and V are subspaces

therefore v+w $\epsilon$ W and v+w $\epsilon$ V

so v+w $\epsilon$ W $\cap$ V

(iii) Now let v $\epsilon$ W $\cap$ V and $\alpha$ be any scalar

v $\epsilon$ W and v$\epsilon$ V

since W and V are subspaces

therefore $\alpha$v $\epsilon$ W and $\alpha$v$\epsilon$V

so, $\alpha$v$\epsilon$ W $\cap$ V

Hence by Subspace Test

W $\cap$ V is a subspace of U. Hence Proved.

Fact used here:-

Subspace Test:-

A subset W of a vector space U is a subspace of U if and only if the following conditions holds:-

(i) W is nonempty. i.e. 0 $\epsilon$ W

(ii) v+w $\epsilon$ W for all v, w $\epsilon$ W

(iii) $\alpha$v $\epsilon$ W for any scalar $\alpha$ and v$\epsilon$W

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