Conditions on overdetermined linear system to be consistent?

[mosawi]

Homework Statement

17) Let A = [1,2 ; -2,-5 ; 2,1 ; 1,-1 ; 1,-2] which is a 5x2 matrix (sorry I don't know how to code a matrix properly).

What conditions must be imposed on vector b for the overdetermined linear system Ax = b (both x and b have arrows on top) to be consistent?

Homework Equations

An overtdetermined system - If m > n, then the linear system Ax = b is inconsistent for at least one vector b in ℝ^n.

The Attempt at a Solution

If m > n (more rows than columns), in which case the column vectors of A cannot span ℝ^m (fewer vectors than the dimension of ℝ^m).

So, there is at least one vector b in ℝ^m that is not in the column space of A, and for that b the system A x = b is inconsistent by Theorem 4.7.1 which states A system of linear equations  is consistent if and only if b is in the column space of A.[STRIKE]So the conditions that must be imposed on b must be to zero out variables? I know this is wrong but I don't really know what to do?[/STRIKE]

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I think I solved it guys, can someone check my solution and tell if its correct if not, what do I need to do to make it correct. I've attached the solution to this poste. Thanks.
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[lippyka]

**The conditions that must be imposed on vector b for the overdetermined linear system Ax = b to be consistent is that b must be in the column space of A. In other words, b must be a linear combination of the column vectors of A. This means that the linear system Ax = b has a solution only if b is in the range of A.