I'm trying to do part (ii) which asks for the basis for the null spaces of the eigenvector equation with the two respective eigenvalues.
What's the easiest way of doing so? I tried to find matrix C manually by letting matrix C be its each individual component (using algebra) and exploiting the fact that the matrix C is symmetric and the two given results for the multiplication of matrix C, but can't seem to progress much.
Edit: Now that I've figured out how to find the basis of the nullspace of the eigenspace corresponding to alpha, how do I find the basis of the nullspace of the eigenspace corresponding to beta? I understand that the basis is one-dimensional.