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 $[E]=[E']$?