I am trying to study lineair algebra on my self, and I came across a weird phenomenom "abusing a notation" What do they mean by that , does that mean that it isnt mathematical correct? Or is it a connection that doesn't really have meaning?
Strictly speaking, if $f(z) = z^k$, then the function is $f$, not $z^k$. But going through the contortions to add another layer of notation would only obscure the plain meaning. So since "we all know" that by $z^k$, we mean the function $z \mapsto z^k$, then let's just write $z^k$.
This depends on what definition for "function" your text actually uses, so I'm guessing a little bit here.
It means I think that it is not technically accurate to describe $z^k$ as a "function", as it is really a polynomial, i.e a formal expression involving powers of $k$. There is a technical distinction to be made between a polynomial and the corresponding function, but they often can be regarded as the same thing. I am not sure how clear my explanation is to you, but basically the comment in the text can be ignored.