In my Danish high school we covered the algebra and geometry of vectors in two and three dimensions in our second year. The high school mathematics curriculum i

In my Danish high school we covered the algebra and geometry of vectors in two and three dimensions in our second year. The high school mathematics curriculum in Denmark is ridiculously unambitious; I'm certain my American friends covered more than this in their high school AP classes. There's a compelling argument for withholding the abstract theory of vector spaces and linear maps from non-mathematicians until multivariate calculus. Linear maps (as approximations to differentiable maps), determinants (as Jacobians measuring volume distortion under pullback), eigenvalues and quadratic forms (as Hessians measuring curvature with eigenvalues as principal curvatures) all make prominent appearances and become easy to motivate geometrically.