Concepts from Complex Vector Analysis and the Fourier Transform

Abstract

In this chapter we present the basic properties of complex vector spaces and the Fourier transform. Sections 1.1 and 1.2 prepare the subject through the standard definitions of linear independence, bases, coordinates, inner product, and norm. In Section 1.3 we introduce linear transformations in vector spaces, emphasizing the conceptual difference between passive and active ones: the former refer to changes in reference coordinates, while the latter imply a “physical” process actually transforming the points of the space. Permutations of reference axes and the Fourier transformation are prime examples of coordinate changes (Section 1.4), while the second-difference operator in particular and self-adjoint operators in general (Section 1.5) will be important in applications. We give, in Section 1.6, the elements of invariance group considerations for a finite N-point lattice. Finally, in Section 1.7 we examine the axes of a transformation and develop the properties of self-adjoint and unitary operators.