A linear combination of some set of vectors, in a space over a field $F$, is arbitrarily defined as the sum of said vectors, which will also be a vector:

$|v\rangle = f_{1}|v_{1}\rangle + f_{2}|v_{2}\rangle + ... + f_{n}|v_{n} = \sum\limits_{i} f_{i}|v_{i}\rangle$, where each $f_{i}$ is some element of $F$. If we have a set of vectors spanning some space, any other vector in that space, can be described by those other vectors.