Linear Algebra - various authors

I realized that I read a lot more than I write about. I know that is hard to believe, and you may be asking yourself how is that possible and does this woman have a job. Yes, I have a job, but right now I have no TV so convert all that time you spend in front of the TV into reading time and what would you get? That doesn’t make my reading any more virtuous than your TV watching, they are both sedentary and I am reading for enjoyment, too. But how many times have you turned on the TV and surfed through those 100+ channels that you get and said to yourself, “There’s nothing interesting on.” Well, one reason I am obsessed with reading is, because there is so much interesting reading that I just haven’t gotten to yet.

Okay, I admit to the obsession. This morning (Sunday morning) I have read parts of two different books and almost an entire magazine, and finally at 1pm I am sitting down to work. I have an article that I want to finish writing today so I can send it off to the co-author. The deadline is May 1, and of course, I am writing a blog instead of working on the article. Sometimes I just have to get the social networking out of the way for the day before I can do my work---but then I do a lot of work.

Reading is also part of my job, and so why don’t I blog about that reading, too. I am a mathematics professor and have been teaching mathematics for 26 years. I am currently on sabbatical, hence the trip to Alaska, and am working on a real-world-project-based linear algebra book. I have been reading a lot of linear algebra and other texts and journal articles in the process. I know that the general reader of my blog, if I had readers at all, are probably not interested in books about linear algebra, but you never know. Here are a few of the books I have found the most useful this semester.

I am looking for applications that don’t appear in the usual linear algebra books, are contemporary, and use techniques that would be in an introductory linear algebra course. One of my colleagues, Heidi Berger, lent me “Parameter Estimation and Inverse Problems” by Richard Aster, Brian Borchers and Clifford H. Thurber. In general, the techniques and applications of this graduate-level text evolved from geophysical inverse methods and does not at first glance sound like a good source, but Heidi knew what she was doing when she lent this book. I think that I want to emphasize the distinction between forward problems (i.e., easy and straightforward) in linear algebra and inverse problems (i.e., needing inverses, eigenvalues and decompositions). This distinction is implicit in linear algebra texts, but I want to make it more explicit and put the reasons for discussing instability, under- and over-determined systems and the need for least squares methods in context.

Moreover, since this book has a geophysical bent it will also help me incorporate my work from before becoming a college teacher into the classroom. During my first career as a computer programmer/analyst I worked for three years in the geophysical exploration area at Phillips Petroleum. In particular, I maintained two huge programs, RayModel and WaveModel, which used rays or waves to model the substructure of the earth. This is an inverse problem: if you collect seismic data from man-made sources, how do you interpret the data to determine the substructure of the earth. The geophysicists use RayModel and WaveModel in a trial-and-error attempt to interpret the data. In RayModel, they described the structure of the earth in a vertical plane and the program ran forward to create a seismograph. They would compare the actual and simulated seismographs, make adjustments to the substructure, and rerun the program. This is one approach to ray-based tomography, but Aster/Borchers/Thurber gave me examples of how to use discretization and linear algebra to determine structures. This idea is used in geophysics as I mentioned, but also in medical and biological technology such as CAT scans. I was able to create a variety of relatively simple problems to use in the classroom from what I learned from this book and other resources on tomography. Sometimes the matrices needed are rather large, but technology such as Maple should make this reasonable for students. We’ll see when I try this in class whether they find it an interesting application or not.

Another useful book has been “Linear Algebra Gems: Assets for Undergraduate Mathematics” edited by David Carlson, Charles R. Johnson, David C. Lay and A. Duane Porter (which makes me wonder where the women are in linear algebra). I have had this book for years and rereading it has been useful. There are many articles in the book that could be assigned to students for independent research, but I found most of the book too theoretical and not based on applications enough to help me with my current work. There are other articles that are useful for the linear algebra instructor, such as how to create matrices that have integral eigenvalues, something that I found useful.

Of course, I started with some idealistic ideas of what I wanted the final product of my sabbatical to look like, and through reading, experimentation and writing, I have built better and more realistic goals for the semester. Of course, the students need to know some theory at the end of the course, although my students will not be taking other courses that use linear algebra per se, so the theory I want them to know is the basis for real-world applications. But, to learn the theory well I think they need to do a lot of experimentation and conjecturing. A good resource for this has been “Linear Algebra with Maple” by William C. Bauldry, Benny Evans and Jerry Johnson. This book was written using Maple V, but the activities are independent of the software (and aren’t we up to Maple 10 now?) A typical exploration is “Describe all matrices that commute with a given 3 by 3 matrix.” I know Benny and Jerry (I actually lived in Benny’s house for a year while he was on leave), and they are always interested in getting the students to think for themselves and to make conjectures based on experimentation, and this book is a good example. I suspect that newer editions have been written or that this group of men (is it always men?) have written other books. I recommend them if you want to use discover-based learning in your class, but again, this book did not help me so much with applications.

Applications do appear in “ATLAST: Computer Exercises for Linear Algebra” by Steven Leon, Eugene Herman and Richard Faulkenberry, but they do not stray far from the typical applications used in standard linear algebra texts. MATLAB is the software the book is written for, but as with the previous book, the exercises and projects can be used with any software. The exercises in this book are interesting, and they provide things you can take into the classroom as well as projects for independent study by students. Of all the books, this is the one I found the most fun, and instead of objectively looking at each chapter for useful information, I find myself going through the exercises to answer the questions posed. I’m easily distracted, and often find I have spent a whole day learning something new and interesting, but that I didn’t make progress toward the goals for the semester.

A book that I keep returning to year after year is “Linear Algebra through its applications” by T. J. Fletcher. Published in 1972, Fletcher did then what I would like to do now, which is to motivate the theory and operations of linear algebra through applications. Many of his applications were applications of linear algebra to areas within mathematics, while I want my applications to be more interdisciplinary in nature, but the idea is the same and I suspect this book has been the inspiration for my work now.

I have read or skimmed dozens of articles and other books, but I won’t begin to list them all. In a small way, this post has helped me organize my thoughts about the reading I have done and will be the beginning of an annotated biography to go in my final sabbatical report. I also cleaned my desk in the process, something that helps me get mentally organized as well. As a last note, I have nothing against men, and historically mathematics has been dominated by men so you would expect most of the linear algebra writing to be by them, but I admit to being somewhat surprised that I have yet to run across a female writer. I can’t help but think there is some glass barrier that I will be butting my head against when I try to publish my work later, and this makes me all the more interested in helping my female students do well in mathematics, along with their male colleagues.