Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum.
Two vector bundles E,E'\to X represent the same class [E]=[E'] in $K(X)$ if there is (after applying some theory) a trivial bundle $K$ such that K\oplus E=K\oplus E'.
What is an example of two non-isomorphic vector bundles $E$, E' such that [E]=[E']?