Discover Linear Algebra

Preface Preface

The purpose of this book is to serve as supporting material for a fairly typical two-semester sequence of courses in Linear Algebra using a discovery-based pedagogical approach.

Each chapter is organized into sections titled Discovery guide, Terminology and notation, Concepts, Examples, and Theory (though some chapters have additional Motivation and/or More examples sections).

The purpose for employing this uniform sectioning scheme is to give the student a uniform flow to encountering each new collection of topics:

Discovery guide

initial encounter through discovery- and problem-solving-based activities;

Terminology and notation

introduction of the communication tools necessary to begin a more sophisticated conversation about the new topics;

Concepts

fuller discussion of the new topics, grounded in reflections on the questions and results of the Discovery guide section;

Examples

computational examples to assist students with the procedural tasks related to the new topics, as well as additional examples that serve to illustrate certain concepts; and

Theory

a more formal and general description of the concepts, with proofs.

Traditional textbooks usually intersperse terminology, concepts, examples, and theory in a linear narrative, and relegate “activities” to the Exercises section at the end of the chapter. By organizing the flow of learning in the above-described manner, it is hoped that the process of encountering and re-encountering (and re-encountering) the topics in different modes — discovery, reflection and discussion, examples, and theory — and at increasing levels of sophistication will lead to deeper learning.

The organization of topics is fairly typical, under the choice of “late vectors” (though the term column vector is used informally in the early chapters). Systems of linear equations are used to motivate matrix theory, up through a basic treatment of determinants and the classical adjoint. Then vectors in \(\R^n\) are introduced as the initial model for how a “vector space” should behave, with an emphasis on a geometric understanding of the vector operations. A basic introduction to abstract vector spaces follows. Next, the topic of matrix forms is used to emphasize how the results of a sustained, careful study of different patterns can be consolidated into a general theory. The treatment is not fully rigorous, as the emphasis (in keeping with the philosophy of this book) is on the journey of discovery rather than on full and uncompromising rigour. (See [7] for more information on the pedagogical philosophy behind this particular unit.) Following this is a standard treatment of abstract inner product spaces, which also contains the basics of matrix adjoints relative to the standard inner products on \(\R^n\) and \(\C^n\text{,}\) as well as a brief discussion of orthogonal/unitary diagonalization. The discussion is kept fairly abstract, so that adjoints of linear operators on inner product spaces could be added as a future topic. And the final topic in this book is a standard treatment of linear transformations, wrapping back around to the topics of similarity and matrix forms at the end by discussing similarity of linear operators.

When using discovery as a pedagogical principal, it is not possible to cover as many topics, or to cover each topic with the same breadth, as in a breakneck-paced lecture class. The goal of these notes is not to teach students a bunch of mathematics in the particular topic of linear algebra, but instead to teach students about mathematics through the discovery of the beautiful and coherent subject of linear algebra. I have tried to distill each topic down to the necessary minimal core of concepts essential to the study of the subject, and have rejected inclusion of peripheral topics and facts or esoteric applications. I do not intend for these notes to be workable for everyone in every kind of linear algebra class. (But since they are released under an open license, they could of course be edited to make them workable for any kind of linear algebra course.)