Proving Linear System Ax=b Consistent iff Rank A = m

[eyehategod]

let A be a mxn matrix.
prove that the system of linear equations Ax=b is consistnet for all column vectors b if and only if the rank of A is m.

I have no idea how to start, can anyone helo me out?

 

[ice109]

what does it mean if the matrix equation is consistent for all vectors b?

 

[eyehategod]

i guess my problem is that i don't quite understand when it says "consistent for all column vectors b."

 

[eyehategod]

also it would mean that b is in the column space of A.

 

[ice109]

also it would mean that b is in the column space of A.

yes but any b?

 

[radou]

i guess my problem is that i don't quite understand when it says "consistent for all column vectors b."

A system of linear equations is consistent if it has a solution. Of course, this solution need not be unique.

 

[HallsofIvy]

A system of linear equations is consistent if it has a solution. Of course, this solution need not be unique.

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.

The OP said earlier, "also it would mean that b is in the column space of A." Okay. And if b is to be any member of A, what must the column space be?