Solving the System Ax = b: Is Full Rank Necessary?

[BobSea]

Homework Statement

The system Ax=b, with Amxn, and m>n, always has a solution when A has full rank. If False, give a counter example, if True, say why.

Homework Equations

None

The Attempt at a Solution

I want to say False because b doesn't need to be in the range of A, so Ax=b wouldn't have a solution. However, I'm having trouble making a counter example (i.e. A = 3x2 matrix and b = 3x1 matrix) that proves the point.

 

[jbunniii]

Your reasoning is correct. For a counterexample, try a very simple A whose columns are elements of the standard basis for $\mathbb{R}^m$.