How to study linear algebra

A first course in linear algebra is dramatically different from most mathematics courses that precede it. The focus shifts from learning computational proceduces to digesting and mastering basic concepts that underlie the computations. To survive, you may need to learn a new way to study mathematics. That's why I wrote this Study Guide - to show you how to succeed in the course and to give you tools to do this.
Because you are likely to use linear algebra later in your career, you need to learn the material at a level that will carry you far beyond the final exam. I believe that the strategies below are crucial to success.

Strategies for success in linear algebra

1. Read each section thoroughly before you begin the exercises. Most students are not used to doing this in courses that precede linear linear algebra. They could survive by looking at the examples only when they were unable to work an exercise. That simply will not work in linear algebra. If you "copy" an example (with necessary modifications), you may think you are "understanding" the problem, but very little true learning will take place.

2. Prepare for each class as you would for a language class. Mastery of the subject requires that you learn a rich vocabulary. Your goal now is to become so familiar with concepts that you can use them easily (and correctly) in conversation and in writing. Pay attention to the warnings here about misuse of terminology.
   This course resembles a language class because of the preparation needed between class meetings to avoid falling behind. Most sections in the text build on preceding sections, and once you are behind, catching up with the class is often difficult. The fact that concepts may seem "simple" does not mean you can afford to postpone your study until the weekend. The homework may be harder than you expect. The most valuable advice I can give you is to keep up with the course.

3. Concentrate more on learning definitions, facts, and concepts, than on practicing routine computations or algorithms. Seek connections between concepts. Many theorems and boxed "facts" describe such connections. My goal is to think in general terms, to imagine typical computations without performing any arithmetic, and to focus on the principles behind the computations.

4. Review frequently. Review and reflection are key ingredients for success in learning the material. I urge you to prepare the sheets as you reach each review box, thinking carefully about the material as you write. Later, you may choose to add further notes and, of course, use the sheets to review for exams. A Glossary Checklist at the end of each chapter may help you learn important definitions.

CAUTION

Because you can find complete solutions here to many exercises, you will be tempted to read the explanations before you really try to write out the solutions yourself. Don't do it! If you merely think a bit about a problem and then check to see if your idea is basically correct, you are likely to overestimate your understanding. Some of my students have done this and miserably failed the first exam. By then the damage was done, and they had great difficulty catching up with the class. Proper use of the Study Guide, however, will help you to succeed and enjoy the couse at the same time.

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