One of many assignments is:
Let $W_1$ and $W_2$ be subspaces of a vector space $V$. Prove that $V$ is the direct sum of $W_1$ and $W_2$ if and oly if each vector in $V$ can be uniquely written as $x_1 + x_2$, where $x_1 \in W_1$ and $x_2 \in W_2$.
I'm not quite sure about the uniquely in the question. Does it actually mean that any $v \in V$ can only be written by one certain pair of vectors from $W_1$ and $W_2$? If so, I don't see why it is necessarily only a one pair of vectors that, added to each other, gives that certain $v \in V$.
Thank you for any explanation !