alt Hacker NewsRe the more abstract approach to transformations, fair point, and I feel like that describes Axler well too. I'd soften my argument to just that being Strang's approach to bringing in the subject.
I agree orthogonality is important. But Strang doesn't get to `a⟂b=0` by means of `cosθ`. You're halfway into the book before he's even defined the Euclidean norm. He derives orthogonality mostly algebraicaly; the only angle he talks about is π/2.
Eh, I don't even think it's more abstract. Like if you're in intro engineering/physics math, and you have R^3 with unit vectors i,j,k, and R^2 with unit vectors I,J, then a function f: R^3 -> R^2 is linear exactly when you can calculate all f(v) = f(ai + bj + ck) = af(i) + bf(j) + cf(k). Then you can define f by "what is f(i)? (Some AI+BJ). What is f(j)? (Some CI+DJ). What is f(k)? (Some EI+FJ)", and then a matrix is just a tabulation of those things. Basically, linear functions let you pick out just n points to define them everywhere, kind of like polynomials. Matrices are that information. Perfectly concrete even at the super intro level. Matrix "multiplication" becomes automatic and trivial; it's just function composition tabulated.
Actually it seems way more concrete to me than mystery rules for "multiplying" arrays.