Rado's theorem

Rado's theorem characterizes exactly which systems of linear equations with integer coefficients have solutions among all integers. Specifically, it states that if you have a set of linear equations, the system has an integer solution whenever the right-hand side vector can be expressed as an integer combination of the system's coefficient vectors, given certain conditions related to the structure of the equations' coefficients. This theorem helps determine when integer solutions exist, connecting the concepts of linear algebra and number theory. It’s fundamental in understanding the solvability of linear Diophantine equations.