Proposition (Criterion for direct sums)

If $U_1,...,U_m$ are subspaces of a vector space $V$, then $U_1+...+U_m$ is a Direct Sum if and only if the only way to write $$0=u_1+...+u_m$$ where $u_i\in U_i$, is by setting $u_j=0, \text{ for } j=1,2,...m$

Proposition (Complement of Subspace Generates Direct Sum)

If $V$ is a finite dimensional Vector Space and $U\subset V$ is a Subspace then there is another subspace $W\subset V$ such that they generate a Direct Sum of the vector space i.e. $$V=U\oplus W$$