My question is:
''In this problem, we shall explore some if the concepts surrounding dual vector spaces for the specific case of $V= P(3)$, the vector space of polynomials with real coeficients of degree less than or equal to $3$. Assume through that $V=P(3)$
a) Describe a basis for $V^*$ (dual vector space) and express the linear functional $f:V\to R$ given by: $$f(a_3x^3+a_2x^2+a_1x+a_0)=2a_3+a_1-5a_0$$ as a linear combination of the basis you found for $V^*$.
b) Find a basis for the annihilator $W$ of the subspace $W= \text{span}(1+x^2, x+x^3)$ which belongs to $C$. What's dimension of annihilator of $W$?
c) We know that the evaluation map $E:V\to V^{**}$ is a vector space isomorphism. Then $E(x^2+x^3)$ is a linear functional on $V^*$. What is the value of the linear functional $E(x^2+x^3)$ on the linear functional $f$ from part (a) ? (I.e, what is $E(x^2+x^3)(f)$?''
Please help me ! I even don't know how to start !
Best Answer
I'll give some hints for you to get started:
If you need further aid ask in comment.