I have found a more "automatic" way to compute a basis for the nullspace of a matrix on Wikipedia, http://en.wikipedia.org/wiki/Nullspace , but from that article, I can see no explanation for why it works. For example, I have no idea as to WHY we are trying to append the identity matrix to the bottom of the matrix to be computed. Also, I do not get why we are trying to get columns whose entries in the top (original) matrix are zeroes. I have tried looking elsewhere, but cannot find anything...
EDIT: I think it works for the same reason that the process of row-augmenting a matrix with the identity matrix, and row-reducing it until we have the identity matrix where the original matrix was works. Heck, by that logic, we could think of the process of column-augmenting our matrix and employing the algorithm talked about in the Wikipedia article as such:
1.) If we were to transpose our augmented matrix, we would have $[A'|I]$, where $A'$ simply means "the transpose of A".
2.) Column-reduction on the column-augmented matrix is equivalent to row reduction on $[A'|I]$; what you will end up with for the column-augmented matrix is $[B|C]'$ (which, of course, means $[B|C]$ for the transpose. It should be noted that $B$ is the row-reduced form of $A'$ (and the transpose is the column-reduced form of $A$). If $A$ has a nullity (dimension of the nullspace) greater than 0, then that means that $A$ is singular, which means that if we were to try to row-reduce the top/left square submatrix of $A$ (or $A'$), we would get a number of rows/columns full of zeroes equal to its nullity. We are interested in these columns, and namely, a record of the operations we did to that submatrix.
I think I have answered my own question, but do not know. So I will leave it here for critique (as my question IS the how/why the algorithm I found works)...
