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In linear algebra, I remember that there was something special about the submatrix in the top right of an rref'd matrix.

1 0 0 |  0 1
0 1 0 | -1 0
0 0 1 |  1 0
-------------
0 0 0    0 0

You would ammend an identity matrix (in this case $2\times2$) above the submatrix and get two vectors in 5 dimensions. I think they say something about the kernel or so, but I do not really remember.

What do those two vectors tell me about the original matrix?

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@Michael: It seems you accidentally entered "spelling" as a tag instead of an edit summary? –  joriki Dec 14 '11 at 17:40

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up vote 2 down vote accepted

Perhaps you mean this: If you negate that submatrix and add the identity matrix below it, you get a basis for the kernel. In your example, you can choose $x_4$ and $x_5$ arbitrarily, and then a general element of the kernel is

$$x_4\pmatrix{0\\1\\-1\\1\\0}+x_5\pmatrix{-1\\0\\0\\0\\1}\;.$$

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Yeah, that is the basis of the kernel that I got out of it per hand. Thanks! –  queueoverflow Dec 14 '11 at 19:05

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