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Math Help - Theory Problem: subspaces of vector space

  1. #1
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    Theory Problem: subspaces of vector space

    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.
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  2. #2
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    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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