I hope the title is not too unclear.
I am given a Matrix $$A\in~\mathbb{R}^{K\times~N},~b\in~\mathbb{R}^{K}$$ and instead of solving the normal equation $min_{x\in~\mathbb{R}^N}|Ax-b|^2_2,~$ an $\alpha\gt0$ is chosen to minimize $$\min\limits_{x\in~\mathbb{R}^N}|Ax-b|^2_2+\alpha|x|^2_2.~(*)$$
The assignment I'm trying to solve is as follows:
Choose an $M\in\mathbb{N}$, a matrix $\tilde{A}\in\mathbb{R}^{M\times~N}$ and a $\tilde{b}\in\mathbb{R}^M$ such that (*) solves $\tilde{A}x=\tilde{b}$.
I recently talked to the professor who gave the assignment and he explained that the solution is to put a $I_N$ below $A$ and therefor add a vector of zeros below $b$, so that $M=K+N$