Let $v_1, v_2,\dots, v_n$ be a basis of the vector space $V$. Let $k$ be an integer such that $1 \leq k \leq n$, and put $U = \operatorname{Span}\{v_1, . . . , v_k\}$ and $W = \operatorname{Span}\{v_{k+1}, . . . , v_n\}$.
I want to show that $V = U\oplus W$.
Would it be enough to prove that $U$ and $W$ are bases of $V$ using linear independence, and then show that there are unique $v\in V$ for every $v = u + w$?