# Question concerning sum of spanning sets

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. – Tunococ Sep 11 '12 at 4:46
Also, $U+W = \mathbb{sp} ( A+B)$. – copper.hat Sep 11 '12 at 5:02
Is that different than what I wrote? i.e. does span(A+B)=span(A)+span(B)? – tk2 Sep 11 '12 at 5:06
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? – Gerry Myerson Sep 11 '12 at 6:19
By the way, there's no such thing as "$A$ is the spanning set of $U$;" rather, $A$ is a spanning set of $U$. – Gerry Myerson Sep 11 '12 at 6:20

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 $U+W$.