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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.

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Looks OK to me. The nullspace and the row space are orthogonal complements to each other, so the nullspace of a matrix whose row space is $U$ will be the row space of a matrix whose nullspace is $U$. –  Gerry Myerson Sep 29 '13 at 1:04
    
Agreed, but is there any way to avoid the part where I generated a bunch of conditions on aij, and then manually used those conditions to construct E, where Ex=0? I thought that was a bit cumbersome, and am wondering if there is a better way to construct E? –  dlaser Sep 29 '13 at 2:01
    
I've posted an answer which you may find to be a slicker way to construct $E$. –  Gerry Myerson Sep 29 '13 at 5:27
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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.

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So the null space of the null space of a matrix, is that matrix's row space? Makes sense; that's what I was missing, this understanding. Thanks! –  dlaser Sep 29 '13 at 21:15
    
That's what I meant when I wrote, "the nullspace and the row space are orthogonal complements to each other" (which I think is a better way to say it, as I am comfortable talking about the nullspace of a matrix, but not the nullspace of a vector space). –  Gerry Myerson Sep 29 '13 at 23:03
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