Elementary Linear Algebra (Textbooks in Mathematics)

Book Preface

In beginning to write a linear algebra text, a question that surfaces even before the first keystroke takes place is who is the audience and what do we want to accomplish. The answer to this is more complex than in several other areas of mathematics because of the breadth of potential users. The book was written with the idea that a typical student would be one who has completed two semesters of calculus but who has not taken courses that emphasize abstract mathematics. The goals of the text are to present the topics that are traditionally covered in a first-level linear algebra course so that the computational methods become deeply ingrained and to make the intuition of the theory as transparent as possible.

Many disciplines, including statistics, economics, environmental science, engineering, and computer science, use linear algebra extensively. The sophistication of the applications of linear algebra in these areas can vary greatly. Students intending to study mathematics at the graduate level, and many others, would benefit from having a second course in linear algebra at the undergraduate level.

Some of the computations that we feel are especially important are matrix computations, solving systems of linear equations, representing a linear transformation in standard bases, finding eigenvectors, and diagonalizing matrices. Of less emphasis are topics such as converting the representation of vectors and linear transformations between nonstandard bases and converting a set of vectors to a basis by expanding or contracting the set. In some cases, the intuition of a proof is more transparent if an example is presented before a theorem is articulated or if the proof of a theorem is given using concrete cases rather than an abstract argument. For example, by using three vectors instead of n vectors.

There are places in Chapters 4 through 7 where there are results that are important because of their applications, but the theory behind the result is time consuming and is more advanced than a typical student in a first exposure would be expected to digest. In such cases, the reader is alerted that the result is given later in the section and omitting the derivation will not compromise the usefulness of the results. Two specific examples of this are the projection matrix and the Gram–Schmidt process. The exercises were designed to span a range from simple computations to fairly direct abstract exercises.

We expect that most users will want to make extensive use of a computer algebra system for computations. While there are several systems available, MATLAB® is the choice of many, and we have included a tutorial for MATLAB in the appendix. Because of the extensive use of the program R by statisticians, a tutorial for that program is also included.