Change of coordinates of the blow-up

Question

I try to compute the change of coordinate of the blow-up, using the notation of 'Principles of algebraic geometry' by Griffiths and Harris on page 184. My problems are in the first steps, defining the inverse of the chart: Let $\tilde{U} = \{ (z,l) \in \mathbb{C}^{n} \times \mathbb{P}^{n-1} \mid z_{i}l_{j} = z_{j}l_{i} \; \forall i,j = 0, \dots, n \}$ and $\tilde{U}_{i} = \tilde{U} \setminus \{ (l_{i} = 0) \}$, then $(\tilde{U}_{i},\varphi_{i})$ is a coordinate subset and the charts are given by $$ \varphi_{i}: \tilde{U}_{i} \rightarrow \mathbb{C}^{n}, (z, l) \mapsto (z(i)_{1}, \dots, z(i)_{i}, \dots, z(i)_{n}) $$ But what is with the inverse? Is it only $$ (t_{1}, \dots, t_{n}) \mapsto ((t_{1}, \dots, t_{n}),l) \; ? $$ And when this is true, the change of coordinates ($i < j$) is $$ \varphi_{j} \circ \varphi_{i}^{-1}(t_{1}, \dots, t_{n}) = \Big(\frac{t_{1}}{t_{j}}, \dots, \frac{t_{i}}{t_{j}}, \dots, \frac{t_{j}}{t_{j}}, \dots, \frac{t_{n}}{t_{j}}\Big) \; ? $$ I'm a little bit irritated because of this notation and I'm not sure with this change of coordinates. Is it possible to write this with the $t_{i}$ or $z(i)_{j}$? Thanks for help.