I am a third year maths student who is currently studying a few modules with applications of linear algebra. I have taken specific linear algebra modules in first and second year. I am completely OK with most of linear algebra, but one thing I hate is writing products of matrices and vectors as summations.
So, for example, if A has dimension nxp, B has dimension pxk, C has dimension kxj, and I wanted to work out column i of (ABC), written as a sum of stuff of ABC it would take me 30 minutes and I would probably get it wrong. My lecturer does this so fluently. Anyone have any tips?
Thanks.
No magic here. I remember being the one that was surprised by my advisor dealing with matrices so quickly, and eventually I got to be on the other side. Still, I'm no "matrix savant".
So, it's all about practice. It's basically about having "row times column" ingrained. After that, it gets to the point where it is about multiplications and additions; I'm definitely not very good at that, so if numbers are not trivial the computation is still painful.
Then there are a few formulas that stay in your mind if you use them enough. I have done enough triple products that I can write $$ (ABC)_{kj}=\sum_{s,t} A_{ks} B_{st} C_{tj} $$ without stopping to think.
For concrete computations, I personally see little value in doing them mentally, as like you say the chances of mistakes are big. Wolfram Alpha multiplies matrices nicely and quickly.