Linear Algebra. Richard C. Penney

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FEATURES OF THE TEXT

      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 right from the beginning, along with the computational. Each abstraction is used to shed light on the computations. In this way, the students see the abstract part of the text as a natural outgrowth of the computational part. This is not the “mention it early but use it late” approach adopted by some texts. Once a concept such as linear independence or spanning is introduced, it becomes part of the vocabulary to be used frequently and repeatedly throughout the rest of the text.

      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 are utilized 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 introduce 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 as they are needed.

      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. With the gradual approach, on the other hand, the question of what a vector space is hardly arises. With this approach, the vector space concept seems to cause little difficulty for the students.

      Treatment of Proofs It is essential that students learn to read and produce proofs. Proofs serve both to validate the results and to explain why they are true. For many students, however, linear algebra is their first proof‐based course. They come to the subject with neither the ability to read proofs nor an appreciation for their importance.

      Many introductory linear algebra texts adopt a formal “definition–theorem–proof” format. In such a treatment, a student who has not yet developed the ability to read abstract mathematics can perceive both the statements of the theorems and their proofs (not to mention the definitions) as meaningless abstractions. They wind up reading only the examples in the hope of finding “patterns” that they can imitate to complete the assignments. In the end, such students wind up only mastering the computational techniques, since this is the only part of the course that has any meaning for them. In essence, we have taught them to be nothing more than slow, inaccurate computers.

      Our point of view is different. This text is meant to be read by the student — all of it! We always work from the concrete to the abstract, never the opposite. We also make full use of geometric reasoning, where appropriate. We try to explain “analytically, algebraically, and geometrically.” We use carefully chosen examples to motivate both the definitions and theorems. Often, the essence of the proof is already contained in the example. Despite this, we give complete and rigorous student readable proofs of most results.

      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.

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