Preface Our goals
This is a textbook for a first-year course in linear algebra. Of course, there are already many fine linear algebra textbooks available. Even if you are reading this one online for free, you should know that there are other free linear algebra textbooks available online. You have choices! So why would you choose this one?
This book arises from my belief that linear algebra, as presented in a traditional undergraduate curriculum, has for too long lived in the shadow of calculus. Many mathematics programs currently require their students to complete at least three semesters of calculus, but only one semester of linear algebra, which often has two semesters of calculus as a prerequisite.
In addition, what linear algebra students encounter is frequently presented in an overly formal way that does not fully represent the range of linear algebraic thinking. Indeed, many programs use a first course in linear algebra as an “introduction to proofs” course. While linear algebra provides an excellent introduction to mathematical reasoning, to only emphasize this aspect of the subject neglects some important student needs.
Of course, linear algebra is based on a set of abstract principles. However, these principles underlie an astonishingly wide range of technology that shapes our society in profound ways. The interplay between these principles and their applications provides a unique opportunity for working with students. First, the consideration of significant real-world problems grounds abstract mathematical thinking in a way that deepens students’ understanding. At the same time, the variety of ways in which these abstract principles may be applied clearly demonstrates for students the power of mathematical abstraction. Linear algebra empowers students to experience what the physicist Eugene Wigner called “the unreasonable effectiveness of mathematics in the natural sciences,” an aspect of mathematics that is both fundamental and mysterious.
Neglecting this experience does not serve our students well. For instance, only about 15% of current mathematics majors will go on to attend graduate school. The remainder are headed for careers that will ask them to use their mathematical training in business, industry, and government. What do these careers look like? Right now, data analytics and data mining, computer graphics, software development, finance, and operations research. These careers depend much more on linear algebra than calculus. In addition to helping students appreciate the profound changes that mathematics has brought to our society, more training in linear algebra will help our students participate in the inevitable developments yet to come.
These thoughts are not uniquely mine nor are they particularly new. The Linear Algebra Curriculum Study Group, a broadly-based group of mathematicians and mathematics educators funded by the National Science Foundation, formed to improve the teaching of linear algebra. In their final report, they wrote
There is a growing concern that the linear algebra curriculum at many schools does not adequately address the needs of the students it attempts to serve. In recent years, demand for linear algebra training has risen in client disciplines such as engineering, computer science, operations research, economics, and statistics. At the same time, hardware and software improvements in computer science have raised the power of linear algebra to solve problems that are orders of magnitude greater than dreamed possible a few decades ago. Yet in many courses, the importance of linear algebra in applied fields is not communicated to students, and the influence of the computer is not felt in the classroom, in the selection of topics covered or in the mode of presentation. Furthermore, an overemphasis on abstraction may overwhelm beginning students to the point where they leave the course with little understanding or mastery of the basic concepts they may need in later courses and their careers.
Furthermore, among their recommendations is this:
We believe that a first course in linear algebra should be taught in a way that reflects its new role as a scientific tool. This implies less emphasis on abstraction and more emphasis on problem solving and motivating applications.
What may be surprising is that this was written in 1993; that is, before the introduction of Google’s PageRank algorithm, before Pixar’s Toy Story, and before the ascendence of what is often called data science or machine learning made these statements only more relevant.
With these thoughts in mind, the aim of this book is to facilitate a fuller, richer experience of linear algebra for all students, which informs the following decisions.
This book is written without the assumption that students have taken a calculus course. In making this decision, I hope that students will gain a more authentic experience of mathematics through linear algebra at an earlier stage of their academic careers.Indeed, a common barrier to student success in calculus is its relatively high prerequisite tower culminating in a course often called “Precalculus”. By contrast, linear algebra begins with much simpler assumptions about our students’ preparation: the expressions studied are linear so that may be manipulated using only the four basic arithmetic operations.The most common explanation I hear for requiring calculus as a prerequisite for linear algebra is that calculus develops in students a beneficial “mathematical maturity.” Given persistent student struggles with calculus, however, it seems just as reasonable to develop students’ abilities to reason mathematically through linear algebra.
- The text includes a number of significant applications of important linear algebraic concepts, such as computer animation, the JPEG compression algorithm, and Google’s PageRank algorithm. In my experience, students find these applications more authentic and compelling than typical applications presented in a calculus class. These applications also provide a strong justification for mathematical abstraction, which can seem unnecessary to beginning students, and demonstrate how mathematics is currently shaping our world.
- Each section begins with a preview activity and includes a number of activities that can be used to facilitate active learning in a classroom. By now, active learning’s effectiveness in helping students develop a deep understanding of important mathematical concepts is beyond dispute. The activities here are designed to reinforce ideas already encountered, motivate the need for upcoming ideas, and help students recognize various manifestations of simple underlying themes. As much as possible, students are asked to develop new ideas and take ownership of them.
- The activities emphasize a broad range of mathematical thinking. Rather than providing the traditional cycle of Definition-Theorem-Proof, Understanding Linear Algebra aims to develop an appreciation of ideas as arising in response to a need that students perceive. Working much as research mathematicians do, students are asked to consider examples that illustrate the importance of key concepts so that definitions arise as natural labels used to identify these concepts. Again using examples as motivation, students are asked to reason mathematically and explain general phenomena they observe, which are then recorded as theorems and propositions. It is not, however, the intention of this book to develop students’ formal proof-writing abilities.
- There are frequent embedded Sage cells that help develop students’ computational proficiency. The impact that linear algebra is having on our society is inextricably tied to the phenomenal increase in computing power witnessed in the last half-century. Indeed, Carl Cowen, former president of the Mathematical Association of America, has said, “No serious application of linear algebra happens without a computer.” This means that an understanding of linear algebra is not complete without an understanding of how linear algebraic ideas are deployed in a computational environment.
- The text aims to leverage geometric intuition to enhance algebraic thinking. In spite of the fact that it may be difficult to visualize phenomena in thousands of dimensions, many linear algebraic concepts may be effectively illustrated in two or three dimensions and the resulting intuition applied more generally. Indeed, this useful interplay between geometry and algebra illustrates another mysterious mathematical connection between seemingly disparate areas.
I hope that Understanding Linear Algebra is useful for you, whether you are a student taking a linear algebra class, someone just interested in self-study, or an instructor seeking out some ideas to use with your students. I would be more than happy to hear your feedback.