Every vector space has a basis. A basis is a linearly independent generating system. So, to even know what a basis is, one must first understand what linear independence and generating system mean. We do this in this chapter. A generating system of a vector space is a set with which it is possible to write every vector of the vector space as a sum of multiples of the elements of the generating system. And linear independence ensures that this representation is unique. In any case, the representation of a vector as a sum of multiples of other vectors is the key to everything: We speak of linear combinations.