What does $Az=y$ and $A^Tx=0$ imply about the relationship between $x$ and $y$?

Question

Suppose A is a $m \times n$ matrix and the vectors $x$ and $y$ are such that $Az=y$ for some vector $z$ and $A^T x=0$. Which one is correct?

1. $x^Ty=0$
2. $||x||_2=||y||_2$
3. $||x||_2 < ||y||_2$
4. $x=ay$ for some real values of $a$

Answer 1

So there's an answer...

(1) is the correct choice.

$x^Ty=x^TAz=x^T(A^T)^Tz=(A^Tx)^Tz=0z=0$