# ece515, homework1 (due 9.11).

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$
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 $$\e$$).

1. 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?
2. 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 $$\e$$.

3. 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?).
4. 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 $$A$$.
5. Find the rank of the matrix $$S$$, where
$(S)_{ij}=i+j.$