Is there a simple way to find a matrix whose null space is the span of a given set of vectors? The problem, and my solution is outlined below. I think my solution is correct, but I feel as if I went about my solution in an awkward way, and that there may be a better/cleaner way to solve the given problem. Would appreciate some feedback on what I did.

 A: I'll take up the story at the point where we have the matrix $$B=\pmatrix{1&0&0&-4&\phantom{-}3\cr0&1&0&-2&\phantom{-}4\cr0&0&1&\phantom{-}0&-1\cr}$$ and we want to find a basis for the nullspace. The non-pivot columns are the 4th and 5th. The nullspace will have a vector with 4th component 1, 5th component zero; by inspection, that vector is $(4,2,0,1,0)$. It will also have a vector with 4th component zero, 5th component 1; by inspection, that's $(-3,-4, 1,0,1)$. So we can take $E$ to be $$\pmatrix{\phantom{-}4&\phantom{-}2&0&1&0\cr-3&-4&1&0&1\cr}$$ 
Note that the rows of my $E$ add up to the 1st row of OP's $E$, while 3 times row 1 plus 4 times row 2 of my $E$ gives 4 times row two of OP's $E$, so both matrices have the same row space. 
More generally, there is a basis for the nullspace containing, for each non-pivot column, one vector where that component is 1 and the other non-pivot components are zero, and the exact value of such vectors is easily read off from the reduced row-echelon form. 
