Effect of permuting coordinates on K-theory Let $A$ be a C*-algebra and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be the linear map which permutes the coordinates via a permutation $\sigma$.  There is an induced map $K_0(C_0(\mathbb{R}^n) \otimes A) \to K_0(C_0(\mathbb{R}^n) \otimes A)$, and it is left as an exercise in several textbooks to show that this map corresponds to multiplication by $(-1)^{|\sigma|}$.  I'm sure this is not difficult, but I'm having trouble; here is my progress.
Since permutations are products of transpositions, we reduce to the case $n = 2$ and $f$ simply swaps the coordinates of $\mathbb{R}^2$.  $f$ is homotopic to the map $g: \mathbb{R}^2 \to \mathbb{R}^2$ given by $g(x,y) = (-x, y)$ via a simple rotation homotopy, so it suffices to show that the map $h: \mathbb{R} \to \mathbb{R}$ given by $h(x) = -x$ induces the inversion automorphism on $K_0(C_0(\mathbb{R}) \otimes A) \cong K_1(A)$.  Sadly, I don't see how to do this.  I'm sure I'll hate myself when I learn how.
Thanks in advance for the help!
Added: I should have mentioned at the outset that I am asking this question to understand a certain proof of the Bott periodicity theorem, and so I'm looking for a fairly elementary argument.

Hint: Using the Künneth Theorem for tensor products (see Blackadars books for instance) it suffices to show that $x\mapsto -x$ induces multiplication by $-1$ on $K_1(C_0(\mathbb R))\cong\mathbb Z$. Now you can just check that a generator is mapped to its inverse.
