Hyper-plane not passing through the orgin
Do you remember the solution to the following equation?
with as a vector and as a scalar. The solution is
Today we see this problem from another view point. Rewrite as
with . So x could be x_1 or x_1+x_0. So x is a hyper-plane! This hyper-plane passes through the point and with normal vector as . Besides among all solutions is the one nearest to the origin. The geometric meaning is obvious.
In addition, if is a matrix with full row-rank, similar results can be obtained.