This seems to be "what are matrices and how do you work with them" and not linear algebra. I mean that can be useful sometimes but seems more like something yo

This seems to be "what are matrices and how do you work with them" and not linear algebra. I mean that can be useful sometimes but seems more like something you would teach in a numerics course instead. Actually, I think this way of explaining and motivating things (linear map==matrix) will get really, really confusing once you try to understand changes of bases or eigenvalue decomposition. A linear map is something that takes vectors and spits out vectors while preserving the vector structure (i.e. addition and scalar multiplication on the input give you addition and scalar multiplication of the output).