## “Matrices in Action”---Reflections on Teaching and Learning in a Changing World

**April 13, 2009**

In 1986, having written a successful textbook in linear algebra and, with George Fix, a book on finite elements, Gilbert Strang made an unusual move: He became an author/publisher. Founded with a very short list (his then new *Introduction to Applied Mathematics*), Wellesley-Cambridge Press has steadily added titles ever since.

An important change for the press came in 2006, when Strang (who was president of SIAM in 1999 and 2000) proposed to SIAM executive director Jim Crowley that SIAM distribute the books together with Wellesley-Cambridge Press. SIAM was happy to make space in the mailroom, and seven Wellesley-Cambridge Press books in mathematics and computational science are now available from both WCP and SIAM.

It seems safe to say that linear algebra is Strang's first love. With the recent appearance (February 2009) of *Introduction to Linear Algebra, Fourth Edition,* he talked to *SIAM News* about that book and other aspects of his multifaceted career.

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*Has the decision to become your own publisher turned out to be a good one? What was your main motivation? Has that goal been achieved? Has publishing your own books shaped the books in any discernible way? How has the agreement with SIAM affected your goals for the books?*Publishing the books has been a wonderful adventure. It is terrific to know where the books go, and to stay in contact with their readers. Writing takes so much time, and I didn't want to send the books away as orphans---who would care for them? I enjoy thinking about the covers too. A constant goal is to present ideas in a clear and lively way.

I am so pleased that SIAM will care for the books too (and that Cambridge University Press will do the same outside North America). I especially approve of keeping the cost reasonable for students.

*Your linear algebra course was one of the first MIT courses to be videotaped in its entirety and made freely available at ocw.mit.edu. In your travels, have you been recognized because of that 18.06 course? More seriously, have you had much feedback on the lectures?*

I feel so fortunate that MIT created OpenCourseWare---giving things freely is the right idea. I was amazed to learn that linear algebra has tied with freshman physics for the most viewers (a million as of last June). E-mail comes all the time about those video lectures. I just heard that the computational science and engineering course (18.085, taped during the fall of 2008) will be added very soon to OCW. (It's on YouTube as of today.)

I encourage everyone to use those lectures and to tape other courses. The world is waiting for high school algebra and calculus, and so much more.

*Since the linear algebra classes were taped, Wellesley-Cambridge Press released the fourth edition of* Introduction to Linear Algebra*. A line illustration on the back of the earlier editions has graduated to the front cover--where it appears in vivid color. What is it with this illustration (shown here), which also appears several times in the book?*That cover picture shows the "Four Fundamental Subspaces"---the column space and the nullspace of A and its transpose. They turn out to be a terrific way to understand a matrix and its rank. The dimensions of the four spaces, their orthogonality, the perfect bases provided by the SVD at the end of the course--so much of linear algebra is in that picture. And students get the idea! My goal is to show matrices in action, and not just sitting there.

Computational Science and Engineering,* another of your textbooks (2007), is leaving SIAM's mailroom in large quantities. Could you tell us how you came to write a textbook in that huge field?*CSE is my other favorite course, taught every year, and it has become quite a large class. The original text was

*Introduction to Applied Mathematics*. The structure of the course has evolved---these days, I start with my favorite matrix (the –1,2,–1 positive definite second difference matrix). And with new engineering examples and more computing, a new book was needed. Allow me to say that the day of simply presenting an infinite series as a solution is over! Engineers compute answers by finite differences and finite elements. We should be teaching the key principles of the methods they use (it is all good mathematics).

The teaching of linear algebra has moved a long way. The course is much more alive, not so formal. It's time for "engineering mathematics" to do the same.

*MIT students can attend their professor's lectures in person, or they can sleep in and watch the taped lectures at their convenience. Has the availability of the lectures on OCW affected attendance in your actual classes?*It's true, the students can learn without coming to class. That happens, and I accept it. I try to put new things into my lectures; Section 1.3 on math.mit.edu/linearalgebra is an example---it tells students very early where the course is going, and points to key ideas like linear independence on which the whole course is built. It is like learning a language; you use words and connect ideas until you get them right.

The good thing for teachers everywhere is that students can review when they are ready (often 3 AM), with no need to wait for faculty office hours.

*Wellesley-Cambridge has also published more specialized books---on wavelets, GPS, and finite elements, always with the same author, but in each case with a co-author. Would you have published these books if you weren't your own publisher? Any future plans along these lines?*You know, every book is special to its authors. My co-authors added so much; they know applications like signal processing and GPS.

I never guessed that my life would be so filled with books---and no regrets. It is so wonderful to connect this way--sharing ideas with friends all over the world.

*Authors often scramble to meet publishers' deadlines. Your writing deadlines are self-imposed, which probably makes them even more pressing. With the publication of Computational Science and Engineering and the fourth edition of Linear Algebra in rapid succession, you must find yourself with unusual amounts of free time. Where have your thoughts turned?*Free time---I wonder what that is. I could mention one crazy problem that popped into my head: Is a random triangle more likely to be acute or obtuse? I was asked to make a video for high school classes, and this will be a perfect topic (as soon as I figure out what "random" means). One answer, based on random angles, is 3 to 1 obtuse. But if I use MATLAB

**rand**10,000 times for the corners, the outputs are about .74 to .26. I've mentioned this question to quite a few people, and I learn something new each time.

Information about SIAM's book program is available at http://www.siam.org/books/. For more about Wellesley-Cambridge Press books, visit http://www.wellesleycambridge.com/.