I wrote this book because I have a deep conviction that mathematics is about ideas, not just formulas and algorithms, and not just theorems and proofs. The text covers the material usually found in a one-semester linear algebra class. It is written however, from the point of view that knowing why is just as important as knowing how.

To ensure that the readers see not only why a given fact is true, but also why it is important, I have included a number of the beautiful applications of linear algebra.

Most of my students seem to like this emphasis. For many, mathematics has always been a body of facts to be blindly accepted and used. The notion that they personally can decide mathematical truth or falsehood comes as a revelation. Promoting this level of understanding is the goal of this text.

R. P.

Parallel Structure: Most linear algebra texts begin with a long, basically computational, unit devoted to solving systems of equations and to matrix algebra and determinants. Students find this fairly easy and even somewhat familiar. But, after a third or more of the class has gone by peacefully, the boom falls. Suddenly the students are asked to absorb abstract concept after abstract concept, one following on the heels of the other. They see little relationship between these concepts and the first part of the course or, for that matter, anything else they have ever studied. By the time the abstractions can be related to the first part of the course, many students are so lost that they neither see nor appreciate the connection.

This text is different. We have adopted a parallel mode of development in which the abstract concepts are introduced only as they are needed to understand the computations. We never introduce a new concept without first justifying its importance and relating it to something already in the students' sphere of experience. In this way, the students see the abstract part of the text as a natural outgrowth of the computational part.

The advantages of this kind of approach are immense. The parallel development allows us to introduce the abstractions at a slower pace, giving students a whole semester to absorb what was formerly compressed into two-thirds of a semester. Students have time to fully absorb each new concept before taking on another. Since the concepts get used as they are introduced, the students see why each concept is necessary. The relation between theory and application is clear and immediate.

Gradual Development of Vector Spaces: One special feature of this text is its treatment of the concept of vector space. Most modern texts tend to introduce this concept fairly late. We put it early because we need it early. Initially, however, we do not develop it in any depth. Rather, we slowly expand the reader's understanding by introducing new ideas and examples as they are needed. Usually, the vector space concepts are introduced in the context of answering a specific question or solving a concrete problem, such as designing a music synthesizer.

This approach has worked extremely well for us. When we used more traditional texts, we found ourselves spending endless amounts of time trying to explain what a vector space is. Students felt bewildered and confused, not seeing any point to what they were learning. the gradual approach, on the other hand, the question of what a vector space is hardly arises. Then the vector space concepts are introduced in a concrete context, the abstractions come across as not only meaningful, but essential.

Applications Fully Integrated into the Text: Teaching a linear algebra class without including a number of the beautiful applications is like writing a mystery novel and leaving out the ending. Our point of view toward applications, once again, is nontraditional. Most linear algebra texts introduce a body of material and afterward provide an example or a "real-world application". We feel that this is backwards. We often begin with the application and show how the necessity to solve a real world problem forces us to develop certain mathematical tools. Not only is this kind of development more interesting for the student, it is also more realistic. Engineers or mathematicians will be required to do exactly this kind of thinking when they get into the workplace.

Conceptual Exercises:Most texts at this level have exercises of two types: proofs and computations. We certainly do have a number of proofs and we definitely have lots of computations. The vast majority of the exercises are, however, "conceptual, but not theoretical." That is, each exercise asks an explicit, concrete question, which requires the student to think conceptually in order to provide an answer. Such questions are both more concrete and more manageable than proofs, and thus are much better at demonstrating the concepts. They do not require that the student already have facility with abstractions. Rather, they act as a bridge between the abstract proofs and the explicit computations.

Attention to Pedagogy: Our style is informal and conversational. We usually begin each section with an example from which we extract a general principle. Most theorems are completely proved. The proofs are often presented, however, as discussions so as to encourage the student to read and understand them.

We also emphasize geometry from beginning to end. We want our students to understand linear algebra "computationally, analytically, and geometrically."

Technology: Most sections of the text include a selection of computer exercises under the heading "On Line." Each exercise is specific to its section and is designcd to support and extend the concepts discussed in that section.

These exercises have a special feature: they are designed to be "free standing." In principle, the instructor should not need to spend any class time at all discussing computing. Everything most students need to know is right there. In the text, the discussion is based on MATLAB. Hovever, translations of the exercises into various other platforms (such as Maple, Mathematica, and TI calculators) are contained in the Student Supplement. They are also available free on the World Wide Web at "http://wiley.com/college."

Glossary and Selected Answers/Hints: The Glossary contains carefully stated definitions of the major concepts in the text. The answer section contains answers to selected exercises as well as hints for others.

Student Supplement: There is a student supplement available which contains more worked out examples, further hints for some of the exercises, and translations of the computer exercises into various platforms (such as Maple, Mathematica, and TI calculators).

Meets LACSG Recommendations: The Linear Algebra Curriculum Study Group (LACSG) recommended that the first class in linear algebra be a "student oriented" class which considers the "client disciplines" and which makes use of tech- nology. The above comments make it clear that this text meets these recommen- dations. The LACSG also recommended that the first class be "matrix oriented." We emphasize matrices throughout.