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At least for me, doing examples and computations is the best way to learn math, and also very important in research. Often an opaque general statement becomes clear after doing a few small examples. In linear algebra, I personally find some of the courses have too few computations, some concepts are best learned by working an hour by hand on some annoying 6 x 6 matrix..

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I like this course by Philip Klein of Brown University for a computational approach: http://cs.brown.edu/courses/cs053/current/index.htm

That said, "working an hour by hand on some annoying 6 x 6 matrix" is how I was taught linear algebra. I got a rare B in the course (even though I aced all the tests!) because there was so much busy work I just refused to do it all. I got literally nothing out of the course in terms of understanding (there was neither time to think nor any real direction given) and a year later I couldn't even do the work anymore. I ended up picking up Axler.

Agreed. I had an additional conflict in that I am a naturally sloppy person, and the worst part of doing matrix calcs by hand is that I'd often make a calculation error due to an inability to read my own handwriting, which would lead to terrible marks again and again.

Learning numpy was godsend.

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