The discussion revolves around the conditions under which the linear system represented by the equation Ax=b is consistent for all column vectors b, specifically exploring the relationship between the rank of matrix A and the consistency of the system. The scope includes theoretical aspects of linear algebra and the implications of matrix rank on solution existence.
Participants generally agree on the definition of consistency in relation to solutions of the system, but there is uncertainty regarding the implications of consistency for all b and the nature of the column space of A.
There are limitations in understanding the implications of the rank of A and the conditions under which the system is consistent for all b, as well as the dependence on definitions of column space and matrix rank.
eyehategod said:also it would mean that b is in the column space of A.
eyehategod said:i guess my problem is that i don't quite understand when it says "consistent for all column vectors b."
A matrix equation, Ax= b, is "consistent" if it has at least one solution. "Ax= b is consistent for all b" means the equation Ax= b is consistent no matter what vector b is.radou said:A system of linear equations is consistent if it has a solution. Of course, this solution need not be unique.