There is a description of relative $K$-theory in terms of complexes of vector bundles. The following is an excerpt from Atiyah's book.
Let $V$ be a complex vector space and consider the exterior algebra $\Lambda^{*}(V)$. We can regard this in a natural way as a complex of vector bundles over $V$. Thus we put $E_{i}=V\times\Lambda^{i}(V)$, and define $V\times\Lambda^{i}(V)\rightarrow V\times\Lambda^{i+1}(V)$ by $(v,w)\rightarrow(v,v\wedge w)$. If $\dim V=1$, the complex has just one map and this is an isomorphism for $v\neq 0$. Thus it defines an element of $K(B(V),S(V))\cong\tilde{K}(S^{2})$ where $B(V),S(V)$ denote the unit ball and unit sphere of $V$ with respect to some metric. Moreover this element is, from its definition, the canonical generator of $\tilde{K}(S^{2})$ except for a sign $-1$.
I don't quite understand the last two statements. Why is it an element of $K(B(V),S(V))$? And where does the sign $-1$ come from?