Preparing to be a linear-algebra teacher: Any tips/suggestions?

Question

I'm engineer and I love linear-algebra. I finished a couple of days the Linear-Algebra on OpenCourseMIT to refresh my memory and I have been doing ton of exercises. I would like to teach to engineer students, so I think it would be a linear-algebra more on the "practical" side.

But I would like to ask you for any tips, suggestions or books that may help me in this process.

For those who teach it, how would you approach it? What are you methods and way?

Anything will be more than welcome.

Thank you.

Answer 1

I have been teaching linear algebra for six years the best book that covers the concepts, mainly for engineers, is Linear Algebra and Its Applications for David C. Lay. And if you want a more slightly abstract book (useful for mathematicians) you can check LINEAR ALGEBRA for Jim Hefferon. The last book I would like to recommend is Linear Algebra Done Right" by Sheldon Axler is an excellent book but I have never used but there is a huge vote for it.

Answer 2

You ask for sources "more on the 'practical' side."

May I suggest consulting Linear Algebra Through Geometry by Thomas Banchoff and John Wermer (Springer link.) You can download each chapter for free.

The figures are dated but still geometric intuition is developed throughout.

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Also I see that MIT Open Courseware has a course video entitled Geometry of Linear Algebra: Link. I haven't looked at it, but from the title it likely similarly emphasizes geometric intuition.

Answer 3

A major problem with most early linear algebra classes is that they emphasize solving problems by hand using rref, as these methods do not translate well to computer calculations since they are unstable and thus can give inaccurate results. While engineering students may not understand the technical definition of stability, they should be taught that

1. Stable algorithms are important for linear algebra on the computer
2. Some simple algorithms (like rref) are good for human calculation, but not stable

With this in mind, when I teach a similar applied course, I plan my lectures around making sure I get as far as eigenvalue decomposition, QR factorization (if possible), and (perhaps most importantly) SVD and at least one good application of each. This allows students to see 2-3 good natural bases for calculations besides just the standard basis, and is enough background for most basic applied linear algebra. Getting as far as SVD is a doable challenge in a one semester course, so the necessary prerequisites determine pretty much everything else.

Strang's linear algebra textbook is one of the best known textbooks that follows this trajectory, but in my opinion it's a bit too advance (with not enough concrete calculations) for a first textbook for engineering students. (It would be an excellent resource for yourself as you're brushing up, as are Strang's own lectures.) Thus I agree with @MrProof and his suggestion of Lay's textbook for students, if you want to follow a textbook closely. It's concrete and has enough calculations to be more accessible for the population you are targeting. It's good to supplement any textbook with 3Blue1Brown's excellent series of videos on linear algebra, which give geometric intuition in a way that no static textbook page will be able to.

Good luck!