Let $A$ be a matrix, and let $B$ be the result of doing a row operation to $A$. Show that the null space of $A$ is the same as the null space of $B$. Let $A$ be a matrix, and let $B$ be the result of doing a row operation to $A$. Show that the null space of $A$ is the same as the null space of $B$.  
Any ideas of what I can do to show this? If the null spaces have to be equal then the vector $x$ in $Ax=0$ and $Bx=0$ have to be equal, but how can I show that $B$ is the result of doing a row operation to $A$ and that $A$ and $B$ are equal?

Doing a row operation on $A$ can be obtained by multiplying $A$ by an invertible matrix $E$. So $B=EA$ for a suitable invertible matrix $E$. If $Av=0$, then $Bv=EAv=0$. Since $A=E^{-1}B$, the converse inclusion also holds.
The matrix $E$ is obtained from the identity $m\times m$ matrix (if $A$ is $m\times n$) by applying the same elementary row operation.
