alt Hacker NewsYou don't need inner products for linear transformations. You just need the idea of a basis and linearity. You define your transformation on a basis (which is all a matrix is: the list of where the map sends each basis element), and it is automatically defined everywhere else via linearity. The textbook my undergrad class used (Curtis) doesn't define inner products until after linear transformations and matrices, for example.
The angular interpretation and geometry are basically the entire point of inner products (inner products are how you define a large chunk of geometry). Angles and projections are the entire intuition behind talking about orthogonality, which is super important practically to basically every field.
Re 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.