A kind of isomorphicity of vector bundles

Question

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?

Note: The above question actualy defines an equivalent relation on the space of all $k$ dimensional subbundles of the $n$ dimensional trivial bundle.

Answer 1

The antipodal map of $S^{2}$ sends $L$ to $L^{-1}$, where $L$ is the line bundle constructed via clutching along an equatorial $S^1$ with the "identity" map $S^1\rightarrow U(1)$.

So let $\alpha$ be the automorphism of $M_{n}(C(S^2))$ induced by the antipodal map, let $e$ be the projection corresponding to $L$, and let $f=\alpha(e)$.