Suppose $T ∈ L(C^3)$is such that 11 and 17 are eigenvalues of T. Furthermore, suppose that T is not diagonalizable. Prove that there exists a vector $(x, y, z) ∈ C^3$ such that: $T(x, y, z) = (55 + 23x, π2 + 23y, √111 + 23z)$.
I've been working through this problem for a while now and cannot figure out where to start. It seems like most of the information is irrelevant except for the fact that we are working with an operator over a three dimensional space that is non diagonalizable and has two guaranteed eigenvalues. I've been trying to prove that the range of this operator is all of $C^3$, but don't know what tools I have to work with. Any hints would be appreciated.