The purpose of this book is to serve as supporting material for a fairly typical introductory course 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;
fuller discussion of the new topics, grounded in reflections on the questions and results of the Discovery guide section;
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
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. Finally, the topic of matrix forms is broached by a simple treatment of eigenvalues/eigenvectors and diagonalization. Note that even though the concept of similar matrices is referenced in this final topic, the topic of “change of basis” has been omitted (see below). However, the full two-semester version of this book does include the topic of change of basis.
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.)Jeremy Sylvestre
Camrose, Alberta 2020