  The Householder algorithm for the QR factorization of a tall thin n x p
full-rank matrix X has the added bonus of producing a matrix M with orthonormal
columns that are a basis for the orthocomplement of the column space of X. We
give a simple formula for M-Transpose x when x is in that orthocomplement. The
formula does not require computing M, it only requires the R factor of a QR
factorization. This is used to get a remarkably simple computable concrete
representation of independent "residuals" in classical linear regression. For
Students problem, when p=1, if R(j)=Y(j)-Ybar are the usual (non-independent)
residuals, W(j)=R(j+1) - R(1)/(sqrt(n)+1) gives n-1 i.i.d. mean-zero normal
variables whose sum of squares is the same as that of the n residuals. Those
properties of this formula can (in hindsight) easily be verified directly,
yielding a new simple and concrete proof of Student's theorem. It also gives a
simple way of generating n-1 exactly mean-zero i.i.d. samples from n samples
with unknown mean. Yiping Cheng exhibited concrete linear combinations of the
Y(j) with these properties, in the context of a constructive proof of Student's
theorem, but that representation is not so simple. Analogous simple results are
obtained for regression when there are more predictors, giving a very simple
computable concrete formula for n-p i.i.d. independent residuals with the same
sum of squares as that of the usual n non-independent residuals. A connection
with Cochran's theorem is discussed.
