Say the null space corresponding to an eigenvalue is the xy plane. For some other eigenvalue, let's assume the null space is the yz plane.
Now, aj for an arbitrary scalar "a" is an eigen vector corresponding to both the eigenvalues. For proving linear independence of eigenvectors for different eigenvalues, we use the Vandermonde determinant after applying the map on a linear combination of the eigenvectors.
But say we chose j as an eigenvector for two eigenvalues, after applying the map, which eigenvalue will we choose? If we take the same eigenvalue twice, the determinant vanishes and hence we cannot prove linear independence.
Some of my peers suggested that I take different eigenvectors, eigenvectors which do not lie in the common set of two null spaces and then proceed, but isn't that what I have to prove in the first place?
PS: sorry I don't know Mathjax