,
calculate
with
(1) A real inner product has four defining properties. What are they?
(2) For the inner product in
,
calculate with
(3) Calculate the cosine of the angle between
and
in the case of (2).
(4) If two vectors are orthogonal with respect to a
particular inner product,
what is their inner product equal to?
(5) Write down the inner product on the vector space of
polynomials 
which
is equivalent to the Euclidean inner product on
.
(6) Write down the inner product on the vector space
of
matrices which
is equivalent to the Euclidean inner product on
.