[zardo]

The car example works as a vector space, but not an inner-product space. There's no sensible way to rotate the length of the car to gasoline in the tank

[javajosh]

That's right! The less pithy answer: objects are the things that stay invariant under a transformation. In that sense then car length and gas are somehow MORE than linearly independent - they are "geometrically independent".

(Note there is a relationship between the length of the car and the gas tank when the car is moving fast (close to c) in your reference frame - but AFAIK there aren't any useful geometric transformations that depend on that fact. :)

[zardo]

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.

[javajosh]

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)

[zardo]

>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.