What I Had Known about Matrix Multiplication Is Just… a Joke

My first encounter with matrix multiplication was early at high school when I read a maths book that I now forgot its title (forgive me, it’s been a very long time ago). There should be no problem for any one with adequate multiplication and addition skill (that’s including me) to understand the numerical formula of it. Though admittedly, I got a hard time to commit that formula into memory due to the fact that it’s easy to mistake columns for rows.

Based on my “understanding”, I can solve numerous maths problems in that book. And because I was “so good” at solving those problems, I never questioned my understanding. Until many years later, matrix multiplication surfaced again in some neural network application. At that moment, I realized that my prior “understanding” is just a joke since I got real trouble numerically interpreting any formula involving matrix multiplication (they are matrixes in 100×100 dimension or more, not friendly 2×2 or 2×3 or 3×3 and it’s just horrible to do the maths the way I did before…).

Honestly, I think it’s not fair to only blame the students, who entered into the track of misconception without ever questioning what was being taught, but the way they were inducted to matrix multiplication has created a whole bunch of confusion and/or illusion (whether by design or not) along the way.

I believe that teaching matrix multiplication to students solely through numerical formula should be deprecated and superseded. Though it’s easy to understand the formula, learners will hardly appreciate the motivation and practical application if they fail to grasp the underlying foundation of matrix multiplication: The linear transformation and all the related definitions that it was built upon. They’re much more beautiful and easier to remember than the hard-cold purely numerical formula.

And for some “hardcore” students the numerical formula sticking into their mind early on could later inflict a huge complication upon their interpretation when some more complex application of matrix multiplication crosses their way. For me, it took a long time to cure myself and unlearn what I’ve learned. Wouldn’t it be better if I didn’t learn it the way I learned it before?