Theory Problem: subspaces of vector space

**antman** · March 15th 2009, 10:04 AM

Let $W_{1}$ and $W_{2}$ be subspaces of a vector space V. Let $W_{1}+W_{2}$ be the set of all vectors v in V such that $v=w_{1}+w_{2}$ , where $w_{1}$ is in $W_{1}$ and $w_{2}$ is in $W_{2}$ . Show that $W_{1}+W_{2}$ is a subspace of V.

**djadooghar** · March 15th 2009, 11:29 PM

It should be easy to show that $W_1+W_2$ is a vector space (please let me know if you need a little help there), from there, we just need to prove that $W_1+W_2\subseteq V$ , assume $v=w_1+w_2\in W_1+W_2$ and since $W_1\subseteq V$ and $W_2\subseteq V$ and $V$ is a vector space, then it particulary is a group, therefore, addition is closed, and so $w_1+w_2\in V$ for any given $w_1\in W_1,\ w_2\in W_2$ , and so we have $W_1+W_2\subseteq V$ .

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