I have only taught "Linear Algebra" once in my life.

It was when Gilbert Strang came out with a wonderful book and I wanted to Learn by Teaching. I had studied linear algebra once at Rice as a sophomore from an extremely old-fashioned text and knew those methods were not a feasible way to get something done.

In high school I had bought a transistor ($15 point contact from Sylvania)and learned about "Thevenin equivalent circuits", so naively for years wondered why I was too stupid to solve the linear equations to build an amplifier. Strang finally let me see I was not stupid, my methods were silly!

I have not forgotten the day in class after I had "shown" (literally) that the product of upper triangular matrices is upper triangular (one easy picture) and some student wondered if we were going to prove it! My answer of "I just did!" was not satisfying to him so I said"If you want me to waste our time by trying to figure out how to put this into symbols so it seems more formal it will not help and there will be deeper Truths we never see!".

From the picture we see that in the easy dot product of one row and another column there is a 0 in every product of elements. It does not'make it more true if we assign row and column numbers, just more seemingly formal.