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Hyper-plane not passing through the orgin

22/08/2011

Do you remember the solution to the following equation?

v^T x =k

with v as a vector and k as a scalar. The solution is

x=v(v^Tv)^{-1}k+x_0

with v^T x_0=0.

Today we see this problem from another view point. Rewrite v^T x =k as

v^T(x-x_1)=0

with x_1=v(v^Tv)^{-1}k. So x could be x_1 or x_1+x_0. So x is a hyper-plane! This hyper-plane passes through the point x_1 and with normal vector as v. Besides among all solutions x_1 is the one nearest to the origin.  The geometric meaning is obvious.

In addition, if v is a matrix with full row-rank, similar results can be obtained.

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