Question

In: Advanced Math

Problem 6. Let Pd (2, C) denote the vector space of C-polynomials in two variables, of...

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).

Expert Solution

(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 .

Colby Messinger answered 1 year ago

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