If $ Ax=b $ has solution what about $ y $ in $ A^Ty=0 $

Question

If $ Ax=b $ has solution what about $ y $ in $ A^Ty=0 $? ($A$ is a $m \times n$ matriz). Do we have them $ y^Tx = 0 $ or $ y^Tb = 0 $ and why ?

Should I explain using only the fact that $ b \in R^m $ and $ x \in R^n $ ? Or there is something more to say?

Answer 1

HINT

We can't conlude in general since we need to distinguish the cases, as for example

• if $m=n$ and $Ax=b$ has solution for every $b$ then $A^Ty=0 \iff y=0$
• if $m=n$ and $Ax=b$ has infinitely many solutions for some $b$ then $A^Ty=0$ has infinitely many solutions
• ...and so on