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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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    $L_2$ norm also called Euclidean Norm2012-02-15

1 Answers 1

2

So there's an answer...

(1) is the correct choice.

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

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    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.2012-02-15