Vector space/Basis complement/Fact

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

u_{1}, \dots, u_{k}

denote linearly independent vectors in $V$ .

Then there exist vectors

u_{k + 1}, \dots, u_{n}

such that

u_{1}, \dots, u_{k}, u_{k + 1}, \dots, u_{n}

form a basis

of

V

.