Vector space/Basis complement/Fact

Let ${}V$ denote a finite-dimensional vector space over a field ${}K$. Let

$u_{1},\ldots ,u_{k}$

denote linearly independent vectors in ${}V$.

Then there exist vectors

$u_{k+1},\ldots ,u_{n}$

such that

$u_{1},\ldots ,u_{k},u_{k+1},\ldots ,u_{n}$

form a basis

of ${}V$.

Write a proof