Let V be a finite dimensional vector space and let A1,A2,...,Ak be subspaces of V such that V = A1 ⊕A2 ⊕···⊕Ak. If the sum from i=1 to k of ai = 0, where each ai is an element of Ai, explain why we can deduce that ai =0 for all i.
How would I go about answering this type of question?
I would use the part of the definition of the direct sum that says $(\sum_{i\neq j}A_i)\cap A_j=\{0\}$.
If $\sum a_i=0$ had a nonzero term $a_j$, you just move the $a_i$, $i\neq j$ over to the right of the equality, and you get the contradiction that both sides are zero.