A kind of isomorphicity of vector bundles

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.

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 $$f=\alpha(e)$$.