Linear Algebra Proof with one-dimensional subspaces Suppose that V is finite dimensional, with $dimV=n$. Prove that there exist one-dimensional subspaces $U_1,...,U_n$ of $V$ such that $$V = U_1 \oplus\dotsb\oplus U_n$$ My linear algebra is rusty, very very rusty. So I'm still in the process of relearning and looking up terms that I will need and/or haven't learned yet. I will most likely ask more questions on any answers or responses I get on here, this potentially could get messy folks...

Since $\dim V=n$ there is a basis $\{v_1,\dotsc,v_n\}$ for $V$. Now put $U_i=\mathrm{Span}\,\{v_i\}$.
