An Intuitive Guide to Linear Algebra
What the author uses as his strawman is a linear algebra course for engineers.<p>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.<p>What the author sees as abstract "(2d vectors! 3d vectors!)" was way more applied than the stuff we dealt in.<p>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.<p>(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.)