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

.