Chapter 2: Linearly independent lists of vectors that span a vector space are of special importance. They provide a bridge between the abstract world of vector spaces and the concrete world of matrices. They permit us to define the dimension of a vector space and motivate the concept of matrix similarity.

Chapter 4 is devoted to several fundamental results of Diophantine geometry such as Siegel's lemma (Lemma 4.1 and Proposition 4.3) and Roth's lemma (Theorem 4.20). Besides them, we also introduce Guass’s lemma, the Mahler measure, the height of a polynomial, Gelfond’s inequality, the index with respect to a weight, the Wronskian, the norm of an invertible sheaf, the height of a norm and the local Eisenstein theorem. We will use them in Chapter 5. Because our purpose is to give a proof of Faltings's theorem in not too many pages, we touch on only the essential results of Diophantine geometry.

We apply the results of the previous chapter to the classical Sturm-Liouville eigenvalue problem, showing that the eigenfunctions form a complete orthonormal basis for L^2. We analyse properties of the solutions of such problems using the Wronskian determinant and define the Green's function that enables us to write an arbitrary solution of the inhomogeneous problem in terms of two particular solutions of the homogeneous problem.