\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\). There exists a natural basis in \(V\), comprised of monomials – we will denote this basis as \(\e\)).
 Is \(V\) a vector space over rational numbers? real numbers? complex numbers? In the cases when it is a vector space, what is its dimension?

Consider the linear function
\[
l:f\mapsto f”(2)
\]
(the value of second derivative of the polynomial \(f\) at \(2\)).Write \(l\) in the monomial basis \(\e\).
 Let \(D:V\to V\) be the operator of differentiation of polynomials. Write the matrix of \(D\) in the basis \(\e\). Find eigenvalue(s?) and eigenvector(s?).
 Let \(S=\{0,1,2,\}\subset \Real\), and \(A:V\to \Fun(S,\Real)\) evaluates the polynomial at the points of \(S\), i.e.
\[
Af=(f(0), f(1), f(2)).
\]
Find the kernel and the mage of \(A\).  Find the rank of the matrix \(S\), where
\[
(S)_{ij}=i+j.
\]
Professor Baryshnikov, I am wondering how I should submit the homework.
login to compass; upload scan to homework 1.