I disagree completely. I am not a physicist but I make video games which has the same kind of constant grounding in applicability. (We are always dealing with running physical systems, it's just that they are simulated.)
The week I learned to treat vectors as abstract objects, rather than arrays of coordinates, I experienced a drastic phase shift in my ability to program geometric operations effectively and clearly. The coordinates are still there, of course, but you have a lot more power over them.
The book "Linear Algebra Done Right" is all about this, and I absolutely recommend reading it if you haven't.
I don't see how we disagree. I wasn't downplaying the importance of the geometric picture, I was just disputing the idea that you could avoid teaching coordinates, or that the coordinate picture was somehow infectious in the sense that it would displace a geometric picture.
I began reading this posting and thinking that I really should acquire material on GA and end up ordering a linear algebra 'refresher" book bases on your comment. Funny considering how difficult it seems for GA to gain mainstream attraction :)
The week I learned to treat vectors as abstract objects, rather than arrays of coordinates, I experienced a drastic phase shift in my ability to program geometric operations effectively and clearly. The coordinates are still there, of course, but you have a lot more power over them.
The book "Linear Algebra Done Right" is all about this, and I absolutely recommend reading it if you haven't.