# Setting some values in a vector to have specific values after multiplying with such matrix

Assume I have a known matrix $$X$$ of size, for example, $$(16$$ x $$16)$$, and a vector $$z$$ of size $$(1$$ x $$16)$$ where only some equispaced values are known, for example $$1:4:16$$ or values in locations $$(1,5,9,13)$$ in $$z$$ are known and others values are unknown. If we have $$y = z*X$$; I need to set the known values in $$z$$ which are located in $$(1,5,9,13)$$ to such numbers such that to have ones in the same locations in $$y$$. Is there any algorithm or method I can use to do that ? Is that possible?