That doesn't work. You're changing the setting of the problem from scalars to vectors in order to avoid breaking the uniqueness properties of fields and rings.

That doesn't work. You're changing the setting of the problem from scalars to vectors in order to avoid breaking the uniqueness properties of fields and rings. The solution set for a system of linear equations is not unique (if it is at all) in the same sense that e.g. additive and multiplicative identity elements are unique.