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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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    This really seems ok to me.2012-09-11
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    Also, $U+W = \mathbb{sp} ( A+B)$.2012-09-11
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    Is that different than what I wrote? i.e. does span(A+B)=span(A)+span(B)?2012-09-11
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    tkrm, you haven't actually found a spanning set, have you? A set $C$, such that the span of $C$ is $U+W$? Think of it this way: everything in $U+W$ is of the form $u+w$ with $u$ in $U$ and $w$ in $W$; then $u$ is a linear combination of elements of $A$, and $w$ is a linear combination of elements of $B$, so $u+w$ is a linear combination of...what?2012-09-11
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    By the way, there's no such thing as "$A$ is **the** spanning set of $U$;" rather, $A$ is **a** spanning set of $U$.2012-09-11
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    In response to your previous comment. It must be that $u+w$ is a linear combination of elements of A and B, correct?2012-09-11
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    Yes. So, what do you get for the spanning set?2012-09-11
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    So for the spanning set you get $\{(u_1+w_1)+...+(u_n+w_n)\}$?2012-09-11
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    Well, 1) you haven't said what $u_1$, $w_1$, etc., are, and 2) whatever they are, once you've finished doing all those additions you've written down, you're left with a set with one element, which is unlikely to be the right answer, and 3) if you want to be sure I see what you write, you have to put in @Gerry. I don't think the Socratic method is working too well here, so I'll post an answer to get you on your way.2012-09-12
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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 $U+W$.