A Householder matrix, or an elementary reflector, has the form Q = I – 2uu^T where u is a unit vector. Show that Q is an orthogonal matrix. Show that Q_v = –v if v is in Span{u} and Qv = v if v is in (Span{u}^⊥. Hense Span{u} is the eigenspace of Q corresponding to the eigenvalue –1 and (Span{u})^⊥ is the eigenspace of Q corresponding to the eigenvalue 1. (Elementary reflectors are often used in computer programs to produce a QR factorization of a matrixA. If A has linearly independent columns, then left-multiplication by a sequence of elementary reflectors can produce an upper triangular matrix.)