# Linear equation Ax=b

1. Oct 19, 2007

### 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?

2. Oct 19, 2007

### ice109

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

3. Oct 19, 2007

### eyehategod

i guess my problem is that i dont quite understand when it says "consistent for all column vectors b."

4. Oct 19, 2007

### eyehategod

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

5. Oct 19, 2007

### ice109

yes but any b?

Last edited: Oct 19, 2007
6. Oct 20, 2007

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

7. Oct 25, 2007

### HallsofIvy

Staff Emeritus
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?