I was reading Zee's QFT book. At the very beginning of the chapter 2 on Dirac equation, he mentions, that Dirac desired for a relativistic wave equation that is linear in spacetime gradient. He goes on to say that for that to happen, a term of the form $c^\mu \partial_\mu$ was required. I understand this. But then he says, had the coefficients $c^\mu$ been numbers, this would form a vector and the term, $c^\mu \partial_\mu$ will no longer be Lorentz invariant.
I understand why $c^\mu$ cannot be just numbers, but, I don't understand why the term would violate Lorentz invariance, had $c^\mu$ been a vector with the components being simple numbers.
What I am looking for is to understand why $c^\mu\partial_\mu$ would violate Lorentz invariance, when $c^\mu$ is vector s formed from simple numbers.
