I’m working through Axler’s Linear Algebra Done Right, 3rd edition. Problem 30 in chapter 5.A asks to prove there exists $x\in R^3$, satisfying $Tx-9x=(-4,5,\sqrt 7)$, if it’s known that -4,5, and $\sqrt 7$ are the eigenvalues of linear transformation T.
My initial reaction to this problem is that each of the three eigenvectors will be sent to scalar multiples of themselves, but my understanding breaks down a little after that. I don’t believe I’m allowed to assume x=c(-4,5,$\sqrt 7$) is an eigenvector, c some constant. Though if that were true, then clearly Tx would be sent to some scalar multiple of x, and the task simply becomes finding what scalar to plug in for c to get back x.
I’m uncertain how to make additional progress and rereading the chapter hasn’t proved fruitful. Am I overlooking some obvious theorem?
Also, I’m new here. Please let me know if I’m violating any rules, and point me toward that list, so that I can avoid breaking any in the future.