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?

Any row operation is achievable by left-multiplying $A$ by some regular matrix. So we have $B=EA$, where $R$ is the (regular) matrix obtained by multiplying some finitely many (regular) matrices corresponding to the elementary row operations. Since multiplying by a regular matrix does not change the rank (there are many possible justifications for this), the result follows.
