By construction every element of \(K(X)\) is a formal difference \([E] \ominus [E']\) where \(E, E'\) are vector bundles over \(X\text{.}\) Such formal differences are sometimes called virtual vector bundles.
Tensor product of vector bundles, \(\otimes \colon \mathrm{Vect}(X) \times \mathrm{Vect}(X) \to \mathrm{Vect}(X)\text{,}\) distributes over \(\oplus\text{.}\) Applying the universal property of Proposition 16.10 in each variable, it therefore extends to a bilinear multiplication on \(K(X)\text{,}\) with \(([E] \ominus [E'])([F] \ominus [F']) = [E \otimes F] \ominus [E' \otimes F] \ominus [E \otimes F'] \oplus [E' \otimes F']\text{.}\) This makes \(K(X)\) a commutative ring with unit \([\varepsilon^1]\text{,}\) and \(f^*\) is a ring homomorphism. In the language of Exercise 16.25, \(\bigl(\mathrm{Vect}(X), \oplus, \otimes\bigr)\) is a commutative semi-ring and \(K(X)\) is its group completion.
If \(X\) is compact Hausdorff, then for every vector bundle \(E \to X\) there is a vector bundle \(E^\perp \to X\) with \(E \oplus E^\perp \cong \varepsilon^n\) trivial (Hatcher, Vector Bundles and K-Theory, Prop. 1.4 for constant rank, and the proof of Prop. 2.1 in general). Consequently every class \([E] \ominus [F] \in K(X)\) can be written in the form \([H] \ominus [\varepsilon^n]\text{,}\) namely with \(H = E \oplus F^\perp\) where \(F \oplus F^\perp \cong \varepsilon^n\text{.}\) Two bundles \(E, E'\) define the same class in \(K(X)\) if and only if they are stably isomorphic, meaning that \(E \oplus \varepsilon^n \cong E' \oplus \varepsilon^n\) for some \(n\text{.}\) Indeed, \([E] = [E']\) means \(E \oplus G \cong E' \oplus G\) for some bundle \(G\text{,}\) and adding \(G^\perp\) to both sides makes \(G\) trivial.
If \(X\) is paracompact and contractible, then by Corollary 16.8 every rank-\(k\) bundle is isomorphic to \(\varepsilon^k\text{,}\) so \(\mathrm{Vect}(X) \cong \mathbb{N}\) and
For \(X = S^2 \cong \mathbb{C}P^1\text{,}\) let \(\gamma\) be the tautological line bundle of Example 16.4. It is not trivial, as we will see in Example 17.8. One can show that \(K(S^2)\) is generated as a ring by \([\gamma]\) subject to the single relation
This is Corollary 2.3 in Hatcher, Vector Bundles and K-Theory, the case \(X = \mathrm{pt}\) of his fundamental product theorem. The relation encodes the isomorphism \((\gamma \otimes \gamma) \oplus \varepsilon^1 \cong \gamma \oplus \gamma\text{,}\) which Hatcher proves with clutching functions in Example 1.13. Additively, \(K(S^2) \cong \mathbb{Z} \oplus \mathbb{Z}\) with basis \([\varepsilon^1]\) and \([\gamma] \ominus [\varepsilon^1]\text{.}\)
