Answer to Question #24834 in Linear Algebra for Matthew Lind

Question

Associated with each equation (Ax)_i=b_i in the system Ax=b there is a hyperplane H_i defined to be the subset of J-dimensional column vectors given by:

H_i = {x|(Ax)_i=b_i}.

Show that the ith column of A^(conjugate transpose) is normal to the hyperplane H_i; that is, it is orthogonal to every vector lying in H_i.

Answer 1

If vector v is normal to H_i = {x|(Ax)_i=b_i} iff it is normal to H ={x|(Ax)_i=0}. But normality followsfrom a_i1 * x1 + ... + a_in * xn = 0, if we take conjugate (a_i1)' * (x1)' +
... + (a_in)' * (xn)' = 0'=0
Last formula means that column ((a_i1)', ... , (a_in)') orthogonal to (x1, ...
,xn), that was necessary.