TITLE: Which vectors have unique representation when it isn't a direct sum?
QUESTION [7 upvotes]: I know that $V = U \oplus W$ means that every $v \in V$ can be written uniquely as $v = u + w$ for some $u \in U, w \in W$. However, what happens if $V = U + W$ is not direct? Then this means that some vector $v$ does not have a unique representation. But can we say exactly which vectors have a unique representation, and which don't?
Is this even a useful question? Intuitively, I am thinking "sometimes we don't have a direct sum, but perhaps we can still work with what we've got. In particular, let's see which vectors we can write uniquely, and maybe we can work with those."

REPLY [11 votes]: If $V=U+W$ is not direct, then there is $z \in U \cap W$ with $z \ne 0.$
Let $v \in V,$ then there are $u \in U$ and $w \in W$ such that
$$v=u+w.$$
We also have
$$v=(u-z)+(w+z).$$
Observe that $u-z \in U, w+z \in W, u-z \ne u$ and $w+z \ne w.$
Consequence: each $v \in V$ does not have a unique representation.