In this chapter we study maps between vector spaces that are compatible with the two vector space operations, addition and scalar multiplication. These maps are called linear maps or homomorphisms. We first investigate their most important properties and then show that in the case of finite dimensional vector spaces every linear map can be represented by a matrix, when bases in the respective spaces have been chosen. If the bases are chosen in a clever way, then we can read off important properties of a linear map from its matrix representation. This central idea will arise frequently in later chapters.