Linear transformations
Richard W. Kaye, Robert Arnott Wilson · 1998
Abstract In this part of the book, we will study linear transformations in the same spirit as we studied inner products and quadratic forms. In this chapter we will see how a linear transformation can be represented by a matrix, with respect to a particular basis, and in later chapters we will discuss how to find the ‘best’ basis, so that the corresponding matrix is as simple as possible, with applications to the solution of many kinds of differential and difference equations, and to quadratic forms. Our vector spaces will be over a field F. For this chapter, F can be any field whatsoever. If it helps, it is possible to think of F as either ℝ. or ℂ without much being lost. In later chapters, we will need some further properties of the field, taking the form that certain polynomials have roots; these properties are always true of the field ℂ, so it is safe to think of F as ℂ throughout the rest of the book.