Recently I supposed to prove the following criterion of the consistency of the linear system:
a linear system $Ax = b$ is consistent iff from $pA = 0$ follows $pb = 0.$ ($p$ is some string of the properly length)
I've successfully proved it, and then I came across the following statement which I supposed to examine whether it's true or not:
the system $Ax = b$ is consistent iff $A^TAx = A^Tb$ is consistent ($A$ is $m \times n$ matrix $n, m \ge2$)
if $Ax = b$ is consistent then $A^TAx = A^Tb$ is obviously consistent. Let's assume the conversly: if $A^TAx = A^Tb$ is consistent then from $pA^TA = 0$ follows $pA^Tb = 0$ or from $qA = 0$ follows $qb = 0$ $(q = pA^T)$ hence $Ax = b$ is consistent and therefore the statement is true. But in my textbook it's said that the statement is false.
Could you please help me to figure out what I've done wrong? Thanks a lot in advance for any help!
For some $q$, there may be no $p$ for which $q=pA^T$ holds.