Consider a linear equation $$ Ax=b,\quad b\in\operatorname{col}(A). $$ The vector $b$ lies in the column space of $A$ so the solution of this linear equation exists.
It is known that any solution has the form $$ x = A^\dagger b + (I-A^\dagger A)h $$ for arbitrary $h$, where $A^\dagger b$ is the least-square solution of $Ax=b$ (as a specific solution), and $(I-A^\dagger A)$ has the same null space with $A$.
Now, consider a weighted least-square problem: $$ \min_{Ax=b} x^\top Rx, $$ where $R$ is positive definite. The solution is $$ x = R^{-\frac{1}{2}}\left(AR^{-\frac{1}{2}}\right)^\dagger b. $$
My question is, by taking arbitrary positive definite matrices $R$, is it possible to obtain all solutions of $Ax=b$, just like taking arbitrary $h$ to obtain all solutions by $x = A^\dagger b + (I-A^\dagger A)h$?
