Blimey. That was a journey... I've always wanted to know more about mathematics in dimensions higher than 3. Turns out I should be careful what I wish for!
Given my background and prior knowledge of 3d transformations, I thought I might find this section rather easy. Though shortly after starting it, I was forced to think about what would constitute an orthonormal basis in four dimensions. Needless to say, this is not only unintuitive, but also impossible to imagine (and as a colleague quite correctly pointed out, impossible to draw!). So instead of relying on intuition, you rely on basic generalised properties of manipulation of bases. After that, you can work in any dimension you want!
Interestingly, amongst all this talk of linear transforms and bases, there was not one mention of cross products. Everything was done without the need for them, which is an eye-opener given the liberal scattering of cross products in my own transformation code!
Perhaps one of the most interesting things I looked at was a description of a basis in four dimensions that you go on to prove is actually representable in three dimensions. Obviously, it's not always possible to do, but this representation of higher dimensions in lower ones was something else I'd been interested in for a while. The 4D window on dimensionality has begun to open!
This section was really fun. It's a shame there is only one section on it this time around. Next up... the dreaded Analysis...