  We show that the orthogonal projection operator onto the range of the adjoint
of a linear operator T can be represented as UT, where U is an invertible
linear operator. Using this representation we obtain a decomposition of a
multivariate Normal random variable Y as the sum of a linear transformation of
Y that is independent of TY and an affine transformation of TY. We then use
this decomposition to prove that the regular conditional distribution of a
multivariate Normal random variable Y given a linear transformation TY is again
a multivariate Normal distribution. This result is equivalent to the well-known
result that given a k-dimensional component of a n-dimensional multivariate
Normal random variable, where k < n, the regular conditional distribution of
the remaining (n - k)-dimensional component is a (n - k)-dimensional
multivariate Normal distribution.
