Linear Algebra with Applications is an introductory text targeted to second or advanced first year undergraduate students. The organization of this text is motivated by what our experience tells us are the essential concepts that students should master in a one semester undergraduate Linear Algebra course. The centerpiece of our philosophy regarding the presentation of the material is that each topic should be fully developed before moving on to the next. In addition, there should be a natural connection between topics. We take great care to meet both of these objectives. This allows us to stay on task so that each topic can be covered with the depth required before progressing to the next logical one. As a result the reader is prepared for each new unit and there is no need to repeat a concept in a subsequent chapter when it is utilized.
Linear Algebra is taken early in an undergraduate curriculum and yet offers the opportunity to introduce the importance of abstraction, not only in mathematics, but in many other areas where Linear Algebra is used. Our approach is to take advantage of this opportunity by presenting abstract vector spaces as early as possible. Throughout the text, we are mindful of the difficulties that students at this level have with abstraction and introduce new concepts first through examples which gently illustrate the idea. To motivate the defini- tion of an abstract vector space, and the subtle concept of linear independence, we use addition and scalar multiplication of vectors in Euclidean Space. We have strived to create a balance between computation, problem solving, and ab- straction. This approach equips students with the necessary skills and problem solving strategies in an abstract setting that allows for a greater understanding and appreciation for the numerous applications of the subject.
The DeFranza & Gagliardi Approach in Their Words
• Our approach is somewhere between the two described above. We cover all the necessary material on systems of equations and determinants in a chapter of approximately 70 pages. This allows us to get to vector space concepts quickly so the instructor can spend most of the semester on linear combinations, vector spaces, linear transformations and core concepts in linear algebra.
• Our examples are selected so the amount of computation is not so long that the reader becomes overwhelmed and discouraged. Yet the examples are chosen to describe the concepts clearly and completely.
• We take every opportunity to recall an argument that the students may have seen in calculus rather than simply referring to the material. Or purpose is to make the connections between linear algebra and the abstraction presented in the course to more concrete topics they have seen already.
• The last chapter will contain applications that use vector space notions, not just applications of systems of equations which are in the second chapter.
• The first chapter covers essential topics needed to continue in higher level mathematics courses. This chapter can be covered quickly or skipped. However, we do not assume every student entering linear algebra has a firm grasp of these essential concepts that are needed in any linear algebra course. With the inclusion of this first chapter, our text can also serve as a bridge course to higher level mathematics courses.
• We use language that is engaging and clear. We understand when simple statements can cause confusion in our students and avoid using terminology that sidesteps explanation. We make connections to what they have seen in other courses whenever possible.
• Some of the organization of our text is driven by our experiences in the classroom. One example is the concept of linear independence. We have included a chapter called Linear Independence, which is not found in other texts. The reason is our experience tells us that most students find this concept confusing. Most texts include this concept as part of a section spending very little time on the topic. We feel it is the basis for many of the topics covered in vector spaces and should be given more importance and time.
• Many authors spend a great deal of time, and many pages, early in the text on the Euclidean spaces, presenting many vector space concepts using this special example. The material is then redone in the context of abstract vector spaces. We think this places the emphasis in the wrong place. The importance of linear algebra, besides the material, comes from the opportunity to introduce the importance of abstraction. We elect to get to abstract vector spaces quickly and use as one set of examples the spaces students are familiar with.
• We are also committed to keeping the text to a reasonable length by being selective with the topics included and sticking to the essential topics needed for a semester course at the undergraduate level.
