The Subject of This Textbook

Before starting with the content of the text, we first ask the basic question: what is linear algebra?

• Linear: having to do with lines, planes, etc.
• Algebra: solving equations involving unknowns.

The name of the textbook highlights an important theme: the synthesis between algebra and geometry. It will be very important to us to understand systems of linear equations both algebraically (writing equations for their solutions) and geometrically (drawing pictures and visualizing).

At the simplest level, solving a system of linear equations is not very hard. You probably learned in high school how to solve a system like

However, in real life one usually has to be more clever.

• Engineers need to solve many, many equations in many, many variables. Here is a tiny example:
• Often it is enough to know some information about the set of solutions, without having to solve the equations in the first place. For instance, does there exist a solution? What does the solution set look like geometrically? Is there still a solution if we change the to a
• Sometimes the coefficients also contain parameters, like the eigenvalue equation
• In data modeling, a system of equations generally does not actually have a solution. In that case, what is the best approximate solution?

Accordingly, this text is organized into three main sections.

1. Solve the matrix equation (chapters 2–4).

   • Solve systems of linear equations using matrices, row reduction, and inverses.
   • Analyze systems of linear equations geometrically using the geometry of solution sets and linear transformations.

2. Solve the matrix equation (chapters 5–6).

   • Solve eigenvalue problems using the characteristic polynomial.
   • Understand the geometry of matrices using similarity, eigenvalues, diagonalization, and complex numbers.

3. Approximately solve the matrix equation (chapter 7).

   • Find best-fit solutions to systems of linear equations that have no actual solution using least-squares approximations.
   • Study the geometry of closest vectors and orthogonal projections.

This text is roughly half computational and half conceptual in nature. The main goal is to present a library of linear algebra tools, and more importantly, to teach a conceptual framework for understanding which tools should be applied in a given context.

The subtle part of the subject lies in understanding what computation to ask the computer to do for you—it is far less important to know how to perform computations that a computer can do better than you anyway.

Uses of Linear Algebra in Engineering

The vast majority of undergraduates at Georgia Tech have to take a course in linear algebra. There is a reason for this:

Here we present some sample problems in science and engineering that require linear algebra to solve.

How to Use This Textbook

There are a number of different categories of ideas that are contained in most sections. They are listed at the top of the section, under Objectives, for easy review. We classify them as follows.

• Recipes: these are algorithms that are generally straightforward (if sometimes tedious), and are usually done by computer in real life. They are nonetheless important to learn and to practice.
• Vocabulary words: forming a conceptual understanding of the subject of linear algebra means being able to communicate much more precisely than in ordinary speech. The vocabulary words have precise definitions, which must be learned and used correctly.
• Essential vocabulary words: these vocabulary words are essential in that they form the essence of the subject of linear algebra. For instance, if you do not know the definition of an eigenvector, then by definition you cannot claim to understand linear algebra.
• Theorems: these describe in a precise way how the objects of interest relate to each other. Knowing which recipe to use in a given situation generally means recognizing which vocabulary words to use to describe the situation, and understanding which theorems apply to that problem.
• Pictures: visualizing the geometry underlying the algebra means interpreting and drawing pictures of the objects involved. The pictures are meant to be a core part of the material in the text: they are not just a pretty add-on.

This textbook is exclusively targeted at Math 1553 at Georgia Tech. As such, it contains exactly the material that is taught in that class; no more, and no less: students in Math 1553 are responsible for understanding all visible content. In the online version some extra material (most examples and proofs, for instance) is hidden, in that one needs to click on a link to reveal it, like this:

Finally, we remark that there are over 140 interactive demos contained in the text, which were created to illustrate the geometry of the topic. Click the “view in a new window” link, and play around with them! You will need a modern browser. Internet Explorer is not a modern browser; try Safari, Chrome, or Firefox. Here is a demo from Section 6.5:

Feedback

Every page of the online version has a link on the bottom for providing feedback. This will take you to the GitHub Issues page for this book. It requires a Georgia Tech login to access.