Linear independence doesn't guarantee that you can rotate, for that you need to know how to exchange one dimension for another.

Essentially you need to define a Pythagorean theorem for your space. You can do that for the car vector space, but there isn't a natural choice.

Isn't that the definition of rotation, the "exchanging one dimension for another"? In 2-D it keeps a point invariant, in 3-D a line, and in N-D it keeps the N-1 object invariant. (Side note: the more basic operation is reflection, since you can get rotation from two reflections)

>Isn't that the definition of rotation, the "exchanging one dimension for another"?

Maybe? I think it allows solutions not typically considered to be rotations, like the special relativity example.