Vector space/Basis complement/Fact/Proof

Let ${\displaystyle {}b_{1},\ldots ,b_{n}}$ be a basis of ${\displaystyle {}V}$. Due to the basis exchange theorem, there are ${\displaystyle {}n-k}$ vectors from the basis ${\displaystyle {}{\mathfrak {b}}}$ that, together with the given vectors ${\displaystyle {}u_{1},\ldots ,u_{k}}$, form a basis of ${\displaystyle {}V}$.