Let's say $\displaystyle u_{1}, u_{2}..., u_{k}$ are linearly independent vectors in $\displaystyle R^n$.

Suppose $\displaystyle u_{k+1}$ is a vector in $\displaystyle R^n$ and $\displaystyle u_{k+1}$ is not a linear combination of $\displaystyle u_{1}, u_{2}..., u_{k}$.

Show that $\displaystyle u_{1}, u_{2}..., u_{k}, u_{k+1}$ are linearly independent.

I have some idea, but I can't think of a rigorous proof. Any ideas?