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
    \[
    (S)_{ij}=i+j.
    \]

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