Suppose that you have $n$ subspaces of a vector space $V$, labeled $U_i$ for $1\leq i\leq n$. When is the sum of these subspaces equal to $V$?
This source seems to imply that if the sum of these subspaces is a direct sum, then the direct sum equals $V$. But from what I am able to gather, the direct sum of subspaces is only equal to $V$ iff the sum is equal to $V$ and the sum is direct.
In any case, how do I determine whether or not a direct sum of subspaces is equal to the containing vector space?