Visual Linear Algebra is a new kind of textbook -- a blend of interactive computer tutorials and traditional text. The computer tutorials provide a lively learning environment in which students are introduced to concepts and methods and where they develop their intuition. The traditional sections constitute the backbone that supports the development of theory and where students' understanding is solidified. Although the design of Visual Linear Algebra is novel, our goals for the book are quite traditional. Foremost among these is to provide a rich set of materials that help students achieve a thorough understanding of the core topics of linear algebra and genuine competence in using them.
The prerequisite for using this textbook in an introductory linear algebra course is one or two semesters of college-level mathematics such as calculus. Having some mathematical maturity is more important than having taken a specific course.
Tutorials and traditional text Visual Linear Algebra covers the topics in a standard one-semester introductory linear algebra course in 47 sections arranged in eight chapters. In each chapter, some sections are written in a traditional textbook style and some are tutorials designed to be worked through using either Maple or Mathematica. Typically, students begin the study of a new group of topics by working through one or more tutorials. Next they progress to a traditional text section that reinforces and further develops what they have just learned. In part, a traditional section provides a summary of the definitions, theorems, and methods that were introduced in the preceding tutorials. It is also where the theory is developed systematically and where most of the proofs are found. Thus, the material of the course is organized in an alternating sequence of interactive tutorials and traditional text sections that complement one another.
About the tutorials Each tutorial is a self-contained treatment of a core topic or application of linear algebra that a student can work through with minimal assistance from an instructor. As in any textbook section, the tutorials have many examples and exercises. In addition, they contain demonstrations, explorations, visualizations, and animations designed to promote a deeper understanding of the material. Each tutorial is, in effect, a section of a linear algebra textbook that has come alive so that a student can actively engage its content.
The 30 tutorials are provided on the accompanying CD both as Maple worksheets and as Mathematica notebooks. They also appear in print as sections of the textbook. In the latter format, the computer commands are replaced by plain English descriptions, and outputs of the commands are displayed. As a result, the printed tutorials are easy for students to read and study away from the computer. Therefore, instructors can choose to treat selected tutorials as traditional text sections rather than have students work through them on a computer.
In the paragraphs below, we describe the ways in which we have worked to attain our overall goal of helping students achieve a thorough understanding of the core topics of linear algebra and genuine competence in using them.
Geometry We use geometry extensively to help students develop their intuition about the concepts of linear algebra. Early in the text we introduce the linear geometry of R2 and R3, and throughout the text we provide a variety of visualizations for students to interact with. As a result, students can gradually develop stronger geometric understanding of the concepts and facts and become more at ease with the general definitions and theorems in higher-dimensional spaces. Furthermore, many of our problems require geometric thinking, which helps students test that their geometric intuition is well grounded.
Applications We have found that students can benefit greatly from working through an application, if the application captures their interest and the materials give them substantial activities that yield worthwhile results. Thus, we developed ten carefully selected applications thoroughly and devoted an entire tutorial to each of them. Since students have the power of a computer at hand, they can carry out realistic computations and can bring the results to a satisfying conclusion. For example, in the study of discrete dynamical systems (Section 7.3), animations of the trajectories permit students to experiment and conduct "what if" analyses on the models presented. In the study of computer graphics (Section 4.5), students create their own animations in 2- and 3-space. In the study of least-squares solutions (Section 8.5), students plot the data and the approximating line or curve and compare the results of different models.
Active learning To encourage students to be active learners, we have designed the tutorials to engage and retain their interest. The exercises, demonstrations, explorations, visualizations, and animations are designed to stimulate students' interest, encourage them to think clearly about the mathematics they are working through, and help them check their comprehension. For example, when students are done with an exercise, they can immediately check their work by opening the closed "Answer" cell that contains a complete solution. Furthermore, since computer commands can be changed and re-executed, students can generate their own examples and can formulate and explore their own questions. At the same time, we want to keep students moving efficiently and productively through the activities to a successful conclusion. Each tutorial, therefore, focuses on the main ideas of the section, poses reasonable challenges, and is designed to be completed in 50 minutes or less.
Active reading The writing style we have employed in this book reflects its student-centered character. Students using this book will typically read and work through a tutorial with minimal assistance from an instructor. Our explanations are therefore more detailed than one might otherwise expect, and our writing style is more direct. As students progress through the course and become more mathematically mature, our writing becomes somewhat less expansive. For example, we explain early proofs in detail, often adding comments about the proof technique or strategy. Later in the book, the proofs are more concise.
Abstraction To help students make the transition to higher mathematics, we have designed our materials so they progress from the concrete and experiential to the abstract and theoretical. In each tutorial, for instance, we start by examining concrete examples and then work toward an understanding of general properties. A similar transition can be seen when the student progresses from a sequence of tutorials to a traditional section, since formal development of theory resides in the latter. Thus, the student acquires an intuitive grasp of the material from working through the tutorials and is thereby prepared for the theoretical developments presented in the traditional section.
Proofs Some linear algebra courses emphasize proofs for an audience of potential mathematics majors; others emphasize computation and applications for an audience of potential science or engineering majors. We have designed Visual Linear Algebra to serve both types of courses. Instructors will find that because the formal development of theory is concentrated in the traditional text sections, students can easily find and review the theory and work through it in a coherent fashion. Since our theory sections are thorough and complete, an instructor teaching a course that emphasizes proofs can choose to assign most of these sections. On the other hand, an instructor whose course emphasizes computation and applications can choose fewer of these sections and can cover just parts of some theory sections. We make it easy for an instructor to assign just parts of these sections, since the early subsections contain the essential parts of the theory, while the later subsections go deeper or cover optional topics.
Problems Linear algebra students are expected to learn many new ways of thinking and to apply them on homework and exam problems. For example, they must learn to use geometric thinking to guide their algebraic manipulations, to apply definitions and theorems with understanding and precision, and to be insightful in choosing from among many possible representations of an object. What we have found most helpful to students in developing these capabilities is having a wide variety of problem types that include many substantial problems. To this end, we provide two complementary categories of problem sets, the computer-based Maple/Mathematica Problems that follow each tutorial and the Pencil and Paper Problems that follow each traditional section and most tutorials. The computer-based problems include explorations, applications, constructions (e.g., of specified types of matrices, figures, or animations), and many challenging computations. Some of the more challenging problems require translating between a geometric and an algebraic representation, uncovering a pattern and justifying it, and using geometric thinking to plan a solution or explain an outcome. The Pencil and Paper Problems include both the familiar types of problems that provide practice using the techniques, definitions, and theory of linear algebra and other types that are less common. Some of the latter check on students' understanding of concepts, some make connections with geometry, and some focus on language. For example, our "Can You Speak Linear Algebra?" exercises help students learn to use the language of linear algebra more precisely and thoughtfully.
We believe that to understand the many important algorithms in linear algebra and to appreciate their implications, students should first work through them, step-by-step, by hand. Thus, we have organized the two categories of problems so that students carry out the fundamental computations by hand, using pencil and paper, before they rely on the computer. For example, students solve linear systems, multiply matrices, evaluate determinants, and find eigenvalues and eigenvectors by hand before they use the computer for the computations.
Technology We have chosen to make substantial use of technology because it provides excellent resources for enhancing student learning. One of our aims in using technology is that students will spend less time at low-level, tedious computations and more time working with core concepts, planning solutions from a higher-level point of view, and carrying out interesting applications relevant to their field of study. Another aim is to give students vivid, dynamic computer visualizations that help them gain greater insights into the underlying geometry of the subject. In addition, the tutorials work well in collaborative learning environments, and they provide a rich set of examples and demonstrations for instructors to use in lectures. On the other hand, we are careful to use technology only where it adds substantial value; where it does not, we stay with the traditional textbook approach.
One challenge in writing a book that depends heavily on a program such
as Maple or Mathematica is to keep the focus on the mathematics,
not on the syntax and minutiae of the software. Our approach has been
to write a complete library of commands (called the VLA library) that
lets students interact with Maple or Mathematica at a more intuitive
level. To aid students in quickly picking up the syntax and usage of the
commands, each command is introduced and explained by means of a simple
example when it is first used.