Lets say you have a linear transformation P. The eigenvalues of the matrices are 0,1 and 2. How would you show that ker P belongs to the eigenspace corresponding to 0? So you have an eigenvalue 0. Let A be the 3X3 matrix. I was thinking of doing something like Ax=λx and substitute 0 for λ. And then show that x,y,z are equal to 0 and hence the eigenspace is 0. Would this be a good idea?