given that

A = L.U
L = \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and U = \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nulspace of A.

• the nulspace of A^T.

I know that the rowspace is just the non zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I cant find a efficient way to calculate basis for nul(A^T). I know that nul(A^T) is the orthagonal projection of col(A), but I still cant manage to find a short easy solution based on some properties without calculating A^T and doing it that way.

If anyone knows some properties to solve this please let me know :)