Prove that if V is a finite-dimensional vector space over a field F, then a subset {$\displaystyle \beta_{1}, \beta_{2}, ..., \beta_{n}$} of V is a basis for V over F if and only if every vector in V can be expresseduniquelyas a linear combination of the $\displaystyle \beta_{i}$.

First things first. Could someone explain the concept of a linear combination? Wikipedia and my book aren't quite cutting it.