Implicit function theorem, what is the meaning of invertible linear operator?

Question

I have to show that around $(1,-1,0)$(have to find the neighborhood as well), $x,y$ are determined uniquely by $z$ given $x+yz-z^3=1, x^3-xz+y^3=0$. What I did so far is: $f:\Bbb{R}^2\times\Bbb{R}\to \Bbb{R}^2$,$f((x,y),z)=(x+yz-z^3,x^3-xz+y^3)$. $f'_x={df\over d(x,y)}=\begin{pmatrix}{1}\\{3x^2-z}\\ \end{pmatrix}$. I know I need to show that $f_x'(a,b)=f'((1,-1),0)=\begin{pmatrix}{1}\\{3}\\ \end{pmatrix}$ and then proceed with the algorithm, except that I've been working on it for too long, and now I don't even know how I am to show or see that a vector $(1,3)$ is a linear operator, and from other computation I actually arrived at linear operators that outputs a vector of vectors and the whole theorem really makes me nervous and unconfident. I could really use your help.