A school psychologist in the United States who was seeking to assess the source of difficulties of a child with learning problems would be very likely to give the child a conventional intelligence test, such as the Stanford-Binet; a French psychologist would be unlikely to use such a test. The reason for the difference is that the two countries have different histories and current traditions with regard to the study and understanding of human intelligence. This particular pair of countries illustrates an especial irony because intelligence testing as we know it began at the turn of the twentieth century in France with a Frenchman, Alfred Binet, whereas widespread use of intelligence testing in the United States did not begin until World War I.

Some fields in psychology and other sciences have a unified history; others do not. Intelligence is one of those fields that does not. For example, French-speaking countries have traditions emanating from Binet and Piaget. English-speaking countries have traditions emanating from Spearman and Thomson (United Kingdom) and Thurstone and Thorndike (United States). German-speaking countries have traditions emanating from Wundt and later, the Gestalt psychologists. Chinese work on intelligence goes back even to before the Common Era, when ability tests were used for selection for jobs. Work in several countries in Africa reveals very different conceptions of intelligence than in Western countries.

Chapter 9 presents an example of how the results of the first eight chapters can be applied to a physical theory – optics. It is all in the nature of applications, and can be omitted without any effect on the understanding of what follows.

Theories of optics

In the history of physics it is often the case that, when an older theory is superseded by a newer one, the older theory retains its validity, either as an approximation to the newer theory, an approximation that is valid for an interesting range of circumstances, or as a special case of the newer theory. Thus Newtonian mechanics can be regarded as an approximation to relativistic mechanics, valid when the velocities that arise are very small in comparison to the velocity of light. Similarly, Newtonian mechanics can be regarded as an approximation to quantum mechanics, valid when the bodies in question are sufficiently large. Kepler's laws of planetary motion are a special case of Newton's laws, valid for the inverse square law of force between two bodies. Kepler's laws can also be regarded as an approximation to the laws of motion derived from Newtonian mechanics when we ignore the effects of the planets on each other's motion.

This book, with apologies for the pretentious title, represents the text of a course we have been teaching at Harvard for the past eight years. The course is aimed at students with an interest in physics who have a good grounding in one-variable calculus. Some prior acquaintance with linear algebra is helpful but not necessary. Most of the students simultaneously take an intensive course in physics and so are able to integrate the material learned here with their physics education. This also is helpful but not necessary. The main topics of the course are the theory and physical application of linear algebra, and of the calculus of several variables, particularly the exterior calculus. Our pedagogical approach follows the ‘spiral method’ wherein we cover the same topic several times at increasing levels of sophistication and range of application, rather than the ‘rectilinear approach’ of strict logical order. There are, we hope, no vicious circles of logical error, but we will frequently develop a special case of a subject, and then return to it for a more general definition and setting only after a broader perspective can be achieved through the introduction of related topics. This makes some demands of patience and faith on the part of the student. But we hope that, at the end, the student is rewarded by a deeper intuitive understanding of the subject as a whole.

In Chapter 10 we go back and prove the basic facts about finite-dimensional vector spaces and their linear transformations. The treatment here is a straightforward generalization, in the main, of the results obtained in the first four chapters in the two-dimensional case. The one new algorithm is that of row reduction. Two important new concepts (somewhat hard to get used to at first) are introduced: those of the dual space and the quotient space. These concepts will prove crucial in what follows.

Introduction

We have worked extensively with two-dimensional vector spaces, but so far always with one of two specific models in mind. A vector space V was either the set of displacements in an affine plane, or it was ℝ2, the set of ordered pairs of real numbers. By introducing coordinates, we were able to identify any two-dimensional vector space with ℝ2 and thereby to represent any linear transformation of the space by a 2 × 2 matrix.

We shall now begin to view more general vector spaces from an abstract and axiomatic point of view. The advantage of this approach is that it will permit us to consider vector spaces that are not defined either in geometrical terms or as n-tuples of real numbers. It will turn out that any such vector space containing only a finite number of linearly independent elements can be identified with ℝn for some integer n so that eventually we shall return to the study of ℝn and the use of matrices to represent linear transformations.