Interactive Linear Algebra (Margalit and Rabinoff)

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).

Overview

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).

The term “algebra” was coined by the the century mathematician Abu Ja’far Muhammad ibn Musa al-Khwarizmi. It comes from the Arabic word al-jebr, meaning reunion of broken parts.

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.