I’m having trouble understanding what is going on in this algorithm. Can someone explain in plain English

- how we build the projection operator?
- what does Linear Algebra have to do this problem? My gloss from reading around on the web is that we are trying to solve a linear system of equations Ax = b where the system is underdetermined. I guess this is why we use the L1 minimization to find the best solution instead of least squares but I haven’t been able to find a good explanation for how this works. Could someone help me out with this?

Thanks.