If your goal is real understanding, the approach that I stated is the right one. If you've learned it the other way and wish to actually understand it, you'll still have to learn the approach that I stated.
Your goal is explicitly not real understanding. And most textbooks do it in the order that you did. (Partly because professors think that most students aren't going to try to understand, so there is no point in giving explanation.) So you probably made the right choice.
But I would still like to see even a passing mention that matrices of just a representation of a linear function given two bases, and matrix multiplication is function composition. That could set a lightbulb on for someone struggling through a problem.
If your goal is real understanding, the approach that I stated is the right one. If you've learned it the other way and wish to actually understand it, you'll still have to learn the approach that I stated.
Your goal is explicitly not real understanding. And most textbooks do it in the order that you did. (Partly because professors think that most students aren't going to try to understand, so there is no point in giving explanation.) So you probably made the right choice.
But I would still like to see even a passing mention that matrices of just a representation of a linear function given two bases, and matrix multiplication is function composition. That could set a lightbulb on for someone struggling through a problem.