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Are the λ-eigenspaces all dimension 1. for each λ1,......,λr.?

Is the dimension of V, r? ie dim(V)=r, ie their contains r elements in a basis for V.

I have another important question, is the matrix A representing the linear transformation T just the diagonal matrix (P

^{-1}AP=D, Where D contains the eigenvalues of T)? Not just in this case, but Always? This ones bugging me.