When I took a course on linear algebra, the professor would introduce the notion of vector spaces and linear transformations first before talking about matrices

When I took a course on linear algebra, the professor would introduce the notion of vector spaces and linear transformations first before talking about matrices, and actually one can derive the matrix multiplication algorithm from the specification of how it must correspond with composition of linear transformations. I thought that the algebraic notation is very concise and appropriate for linear algebra. While new notation like this is interesting, I do wonder how useful it is from a pedagogical standpoint for linear algebra. Category theory OTOH is one of the areas of mathematics with diagrams rule over equations because a single diagram can encode so many equations. For excellent introductions to linear algebra I recommend Linear Algebra Done Right[0], which Springer offered for free for a while (it doesn't seem to be anymore though, contact me if you want the PDF). [0] http://linear.axler.net/ http://linear.axler.net/