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Thread: Theory Problem: subspaces of vector space

  1. #1
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    Theory Problem: subspaces of vector space

    Let $\displaystyle W_{1} $ and $\displaystyle W_{2}$ be subspaces of a vector space V. Let $\displaystyle W_{1}+W_{2} $ be the set of all vectors v in V such that $\displaystyle v=w_{1}+w_{2}$, where $\displaystyle w_{1} $is in $\displaystyle W_{1}$ and $\displaystyle w_{2}$ is in $\displaystyle W_{2}$. Show that $\displaystyle W_{1}+W_{2}$ is a subspace of V.
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  2. #2
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    It should be easy to show that $\displaystyle W_1+W_2$ is a vector space (please let me know if you need a little help there), from there, we just need to prove that $\displaystyle W_1+W_2\subseteq V$, assume $\displaystyle v=w_1+w_2\in W_1+W_2$ and since $\displaystyle W_1\subseteq V $ and $\displaystyle W_2\subseteq V$ and $\displaystyle V$ is a vector space, then it particulary is a group, therefore, addition is closed, and so $\displaystyle w_1+w_2\in V$ for any given $\displaystyle w_1\in W_1,\ w_2\in W_2$, and so we have $\displaystyle W_1+W_2\subseteq V$.
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