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If $U$ and $W$ are subspaces of $V$, a vector space, and $A$ is the spanning set of $U$ and $B$ is the spanning set of $W$. Find the spanning set of $U+W$, in terms of $A$ and $B$, and prove that this is the spanning set. Note that we have seen that $U+W$ is a subspace of $V$.

So from the definition of $\operatorname{span}$, I have that $\operatorname{span}(A)=U$ and $\operatorname{span}(B)=W$. Then $U+W=\operatorname{span}(A)+\operatorname{span}(B)$. This should be true since all elements in $A$ and $B$ are in $U$ and $W$, and by the sum of subspaces. I'm not sure whether I am thinking about that right. Any help is appreciated and thanks in advance.

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    I understand. Thank You.2012-09-12

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Try to prove that if $U$ and $W$ are subspaces of $V$, and $A$ and $B$ are spanning sets for $U$ and $W$, respectively, then the union of $A$ and $B$ is a spanning set for $U+W$.