Vector space/Finite dimensional/Linear subspace/Direct complement/Fact/Proof

Proof

Let $v_1,\dots ,v_k$ denote a basis of $U$. We can extend this basis, according to fact, to a basis $v_1,\dots ,v_k,v_{k+1},\dots ,v_n$ of $V$. Then

$W=⟨v_{k+1},\dots ,v_n⟩$

fulfills all the properties of a direct sum.

To fact