  A short and elementary proof of the joint convexity of relative entropy is
presented, using nothing beyond linear algebra. The key ingredients are an
easily verified integral representation and the strategy used to prove the
  Cauchy-Schwarz inequality in elementary courses. Several consequences are
proved in a way which allow an elementary proof of strong subadditivity in a
few more lines. Some expository material on Schwarz inequalities for operators
and the Holevo bound for partial measurements is also included.
