As an electrical-engineer turned machine-learning-grad-student, linear transformations have been involved in most everything I do since my first year of undergrad. But all this time, I've done matrix multiplication the way I was taught in high school: "The (i,j) element of AB is what you get by walking right across the i'th row of A while you walk down the j'th column of B, taking the sum of products as you go."
It works, but there's no connection between that process and the intuition of a linear transformation; it's just a rote computation. And checking a long string of matrix multiplications to see if they intuitively make sense (shouldn't everything intuitively make sense?) is especially aggravating when you constantly have to interrupt your intuition to switch to a rote calculation.
I never thought to think of the columns of B as vectors that physically travel through A; to think of a dataflow or pipeline from right to left on the page. Sure, it's not a cure-all, but it'll be a useful mental tool to have.
Oh, and it's also an excellent introduction to the subject, although the Linear Operations section gets a bit muddled... first something's not a linear operation, and then it is, wat? Still, an excellent post.
You might get some mileage out of thinking of a row in the matrix product as a dot product, with the intuition that goes along with that. The dot product is perhaps easier to tie into the geometric intuition you have for linear transformations.
It works, but there's no connection between that process and the intuition of a linear transformation; it's just a rote computation. And checking a long string of matrix multiplications to see if they intuitively make sense (shouldn't everything intuitively make sense?) is especially aggravating when you constantly have to interrupt your intuition to switch to a rote calculation.
I never thought to think of the columns of B as vectors that physically travel through A; to think of a dataflow or pipeline from right to left on the page. Sure, it's not a cure-all, but it'll be a useful mental tool to have.
Oh, and it's also an excellent introduction to the subject, although the Linear Operations section gets a bit muddled... first something's not a linear operation, and then it is, wat? Still, an excellent post.