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

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 $a$
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I couldn't find a title for the question! –  Gigili Feb 15 '12 at 19:33
Write out $x^Ty$ using $Az=y$ and then remember that $(X^T)^T=X$ and $(XY)^T=Y^TX^T$ –  Bill Cook Feb 15 '12 at 19:34
How about, for the title, "What does $Az=y$ and $A^Tx=0$ imply about the relationship between $x$ and $y$?" –  David Mitra Feb 15 '12 at 19:40
@BillCook: $x^Ty=x^T(Az)=((Az)^T x)^T=(z^T A^T x)^T$, what should I do next? –  Gigili Feb 15 '12 at 19:41
Observe $(z^T A^Tx)^T=(z^T 0)^T=0$ :) –  David Mitra Feb 15 '12 at 19:46

$x^Ty=x^TAz=x^T(A^T)^Tz=(A^Tx)^Tz=0z=0$
All the rest of the parts can hold by picking the right choice of $A$, $z$, and $x$. They can also fail by picking the "wrong" choice. (1) is the only part which always holds. –  Bill Cook Feb 15 '12 at 20:40