Let $V$ be a [[vector space]] of [[dimension of a vector space|dimension]] $n$. Let $\mathbb{Z}^+\ni r<n$, and let $v_{1}, \ldots, v_{r}$ be [[linearly independent vectors]] of $V$. Then we can find [[Element of a Set|elements]] $v_{r+1}, \ldots, v_{n}$ such that $ \left\{v_{1}, \ldots, v_{n}\right\} $ Is a [[basis for vector space|basis]] of $V$ #### Proof Since $r<n$ we know that $\left\{v_{1}, \ldots, v_{r}\right\}$ cannot form a basis of $V$, and thus cannot be a maximal set of linearly independent elements of $V .$ In particular, we can find $v_{r+1}$ in $V$ such that $ v_{1}, \ldots, v_{r+1} $ are linearly independent. If $r+1<n$, we can repeat the argument. Apply induction until we reach the [[maximal subset of independent vectors|maximal]] $\left\{v_{1}, \ldots, v_{n}\right\}$. And [[maximal subsets form a basis]]