Hello /sci/

I've been stuck in this shit for 2 days with no hope in sight.

In short this is about spectral methods, particularly the TAU spectral method.

I'm trying to solve $\Delta f = 0$ with boundary conditions $f(-1)=0$ and $f(1)=1$ in [-1,1] with Python

Using Tchebyshev discretization with $N=51$ points.

Before asking me to use the TAU method to solve the equation, it asked me to find out the passage matrix from the physical to spectral and vice versa, as well as finding the Tchebyshev differentiation matrix in the spectral domain.

How is this related to solving the equation? Aren't I supposed to formulate $A x = B$ then inverse A to find out x? Please help.