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Suppose I am quite easily able to solve a linear system, $\bf{A}x = b$.

However, what I'm really after is the solution to $\bf{A}^Ty = b$ (notice that it's the same right hand side vector in both cases).

Also, suppose that the latter system, for whatever reason, is much harder / slower for me to solve, and that's why I choose to solve the former system (but a solution exists in both cases).

To make things worse, suppose that I'm using an iterative solver, where I don't ever actually construct the matrix $\bf{A}$, and thus I never have access to it. All I have is $\bf{x}$ and $\bf{b}$.

Is there any way for me to obtain the solution $\bf{y}$, knowing just the above?

Thanks!

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    $\begingroup$ There is no connection between the solutions of the former and the solutions of the latter (unless of course if matrix $A$ is symmetrical !) $\endgroup$
    – Jean Marie
    Commented Oct 3, 2017 at 20:40

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