Problem 1
Let
be a subspace of a finite-dimensional inner product space
.
Show that
, that is, the orthogonal complement
of orthogonal complement of a subspace is the subspace itself.
Problem 2
Describe explicitly all inner products on
and on
.
Problem 3
a) Let
be the linear operator on
defined by
(that is,
is rotation by
).
Find all inner products
on
such
that
for all
. (
Hint: start
by noticing that the standard inner product
satisfies
this condition).
b) Let
be the standard inner product on
.
Prove that there is no non-zero linear operator on
such
that
for all
.