An Intuitive Guide to Linear Algebra
What the author uses as his strawman is a linear algebra course for engineers.As mathematicians, we didn't do any of this matrix and vector stuff with numbers when introducing linear algebra in university.  There were a bunch of axioms, and you proved things.  That's how you know.What the author sees as abstract "(2d vectors! 3d vectors!)" was way more applied than the stuff we dealt in.But, granted, the purpose wasn't learning about how to get mini-spreadsheets for equations.  It was about how to rigorously navigate a useful axiomatic setting.(Later on, we proved that you can find a base, and write down your linear transformation as a bunch of numbers and call that a matrix; same for points and vectors.  But we always saw that as somewhat ugly, and anyway limited to the finite dimensional case.)