> That's exactly what they do though. > and internalizing this is extremely important for beginning math students. I'm not saying that you're wrong, because,

> That's exactly what they do though. > and internalizing this is extremely important for beginning math students. I'm not saying that you're wrong, because, well, you aren't. But does the choice of notation help here? Understanding that a matrix isn't a linear transformation but only a representation of it is something that's inherently hard and requires a certain level of mathematical maturity. At that point in a sense notation doesn't really matter any longer.