# PROOF sUBpace problem

• Jan 28th 2007, 11:38 AM
ruprotein
PROOF sUBpace problem
can some one solve this ?

Let W1 and W2 be subspaces of a vector space V. Prove that W1 union W2 is a subspace of V iff one of the subspaces contains the other.
• Jan 28th 2007, 01:10 PM
ThePerfectHacker
Quote:

Originally Posted by ruprotein
can some one solve this ?

Let W1 and W2 be subspaces of a vector space V. Prove that W1 union W2 is a subspace of V iff one of the subspaces contains the other.

The only if part is trivial. Here http://www.mathhelpforum.com/math-he...oup-proof.html, try to understand it. The proof to your problem is almost exactly the same.
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An alternate way to think of it is like this. \$\displaystyle W_1,W_2\$ are subspaces, thus, they are abelian groups under vector addition (definition of vector spaces). Thus, by the conditions of the problem \$\displaystyle W_1
\cup W_2\$ is a subspace and hence an abelian group. The link I gave you says, the union of two subgroups is a subgroup if and only if one is contained in the other. Now since \$\displaystyle W_1
\cup W_2\$ is an abelian group under addition of vectors follows that one of them is contained in the other. Thus, \$\displaystyle W_1\subseteq W_2\$ or \$\displaystyle W_2\subseteq W_1\$
• Jan 28th 2007, 02:39 PM
Plato
You can also see a proof here.