  A system of linear equations is normally understood as a linear mapping
between two vector spaces. However, most direct solutions (e.g., QR, LU, ...)
rely on the inelegant approach of back-substitution: a significant departure
from such a characterization. In this paper, two new methods are introduced
which respect the underlying vector space throughout, whether it be the row or
column space of the coefficient matrix. Both solutions produce a generalized
inverse as well as a projection operator for the null space. They also have the
unique feature of admitting an online solution, which allows a solution to be
computed as the data becomes available.
