Dimension of the null space of A transpose

1. Oct 26, 2012

leroyjenkens

So I'm given a matrix A that is already in RREF and I'm supposed to find the null space of its transpose.
So I transpose it. Do I RREF the transpose of it? Because if I transpose a matrix that's already in RREF, it's no longer in RREF. But if I RREF the transpose, it gives me a matrix with 2 leading entries that are both equal to zero. I don't think I'm doing it right.

Thanks.

2. Oct 26, 2012

mathwonk

you don't need to do anything to find the dimension of the nullspace of the transpose if you already understand the rank of the matrix, since the nullspace of the transpose is the orthogonal complement of the range of the matrix. so if an nbym matrix represents a map R^m-->R^n of rank r, then the range has dimension r, so its orthocomplement has dimension n-r.

Equivalently, the rank of the transpose is also r. I.e. RREF shows you that both the row rank and the column rank are equal. The rank of a matrix is by definition the column rank, but the columns of the transpose equal the rows of the original matrix.

I.e. just knowing the rank of the original matrix, gives you also the rank of the transpose and hence also its nullity.