In: Advanced Math
Problem 6. Let Pd (2, C) denote the vector space of C-polynomials in two variables, of degree ≤ d.
Define a linear map S : P2(2, C) → P2(C) by S(p) := p(z, z) (where z is a variable for the polynomials in P2(C)).
(a) Prove that S is surjective and that Skew2(2, C) ⊂ ker(S).
(b) Give an example of a polynomial in ker(S) \ Skew2(2, C). Hence write down a basis for ker(S).
(a) In one of the previous part we have seen that a basis of
is
. Also,
we know that a basis of
is
, and a
basis of
is
. Now, by definition of the linear map
, we have
Therefore, the image of
contains the basis
of
,
showing that the image is
itself. Hence,
is surjective.
Since
and
, we conclude that the basis
of the
subspace
is contained in the kernel. But then, the subspace itself must be
contained in the kernel, because kernel is a subspace and hence is
closed under linear span. Thus,
.
(b) Since
and
, by rank-nullity theorem we get
But
gives
, and we already know that
and
is a
basis for
. Thus, if
then a basis for
would be
.
Note that if
then
; thus,
satisfies
, and a basis of
is
.