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Let $A$ and $B$ be subspaces of a vector space $V$. Their sum $A+B$ is defined as:

$$A+B= \{x+y \mid x\in A,y\in B\}$$

It's given that $A+B$ is also a subspace of $V$.

Show that if $X$ is a spanning set for $A$ and $Y$ is a spanning set for $B$ then $X\cup Y$ is a spanning set of $A+B$.

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Hint: $v \in A + B\iff v = a +b$ for some $a \in A$ and $b \in B$. How can we write $a + b$ in terms of elements of $X \cup Y$?

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