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Given an $m\times n$ matrix $\mathbb{A}$, the set of $n$-vectors $\mathbf{x}$ that satisfy $\mathbb{A}\cdot\mathbf{x}=0$ is the null row-space of $\mathbb{A}$.

The elementary row operations on $\mathbb{A}$ are: summing one row to another row and multiply a row by a nonzero constant.

The question is: For all $m\times n$ matrices $\mathbb{B}$ that have the same null row-space as $\mathbb{A}$ (that is, $\mathbb{A}\cdot\mathbf{x}=0$ iff $\mathbb{B}\cdot\mathbf{x}=0$), can $\mathbb{B}$ be obtained from $\mathbb{A}$ by elementary row operations on $\mathbb{A}$?

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Hint As the row operations can be reversed, it is enough to prove/disprove the result in the case $A,B$ are Reduced Row Echelon Forms (RREF).

Moreover, given two RREF, one can be obtained from the other by row operations if and only if they are the same.

Therefore, the problem reduces to: If $A,B$ are RREF, is it true that they have the same Null space if and only if they are equal?

And this is easy to solve if you consider how you get the null space from an RREF.

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  • $\begingroup$ how do you get the nullspace from an RREF? $\endgroup$ – becko May 22 '14 at 21:54
  • $\begingroup$ @becko Do you know how to solve a system of equations by row reduction? $\endgroup$ – N. S. May 22 '14 at 22:01

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