Let a system of vectors $\displaystyle \bold{v}_{1},\bold{v}_{2}, \ldots, \bold{v}_{r} $ be linearly independent but not generating. Show that it is possible to find a vector $\displaystyle \bold{v}_{r+1} $ such that the system $\displaystyle \bold{v}_{1}, \bold{v}_{2}, \ldots, \bold{v}_{r}, \bold{v}_{r+1} $ is linearly independent.

So let $\displaystyle \bold{v}_{r+1} $ be any vector that cannot be represented as a linear combination $\displaystyle \sum_{k=1}^{r} \alpha_{k} \bold{v}_{k} $. So $\displaystyle \bold{v}_{1}, \bold{v}_{2}, \ldots, \bold{v}_{r}, \bold{v}_{r+1} $ does not form a basis for some vector space $\displaystyle V $.

To show linear independence, we have to show that $\displaystyle \sum_{i=1}^{k+1} \alpha_{k} \bold{v}_{k} = \bold{0} $ with $\displaystyle \alpha_{k} = 0 $.