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\begin{document}
\centerline{Math 201 Homework for Friday, Week 6}
\bigskip
Due: Friday, October 8.

\begin{prob}
  Let~$\mathcal{P}_n(\R)$ be the vector space of polynomials in~$x$ of degree at
  most~$n$ with coefficients in~$\R$. Define
  \begin{align*}
    f\colon\mathcal{P}_2(\R)&\to\mathcal{P}_3(\R)\\[8pt]
    p(x)&\mapsto \int_{0}^x p(t)\,dt.
  \end{align*}
  Note that this is a definite integral, so there is no constant of integration! One can show $f$ is a linear transformation (but you don't need to do that for this problem). 
  \begin{enumerate}
    \item Find the matrix representing~$f$ with respect to the ordered bases~$\langle 1,x,x^2
      \rangle$ for~$\mathcal{P}_2(\R)$ and $\langle 1,x,x^2,x^3 \rangle$
      for~$\mathcal{P}_3(\R)$.
    \item Find the matrix representing~$f$ with respect to the ordered bases 
      $\langle 1+x+x^2, x+x^2, x^2
      \rangle$ for~$\mathcal{P}_2(\R)$ and $\langle 1+x+x^2+x^3,x+x^2+x^3, x^2+x^3,x^3 \rangle$
      for~$\mathcal{P}_3(\R)$.
  \end{enumerate}
  \end{prob}
 
  \begin{prob}
    Let $V$ be a finite-dimensional vector space over $F$, and let $f\colon V \to V$ be a linear transformation. Prove that $f$ is one-to-one if and only if it is onto.
  \end{prob}

  \begin{prob}
    Let~$\mathcal{P}(\R)$ be the vector space of polynomials in~$x$ with coefficients in~$\R$. Define
     \begin{align*}
       f\colon\mathcal{P}(\R)&\to\mathcal{P}(\R)\\[8pt]
       p(x)&\mapsto \int_{0}^x p(t)\,dt.
     \end{align*}
     and
         \begin{align*}
       g\colon\mathcal{P}(\R)&\to\mathcal{P}(\R)\\[8pt]
       p(x)&\mapsto p'(x).
     \end{align*}
     Again, one can show that $f$ and $g$ are linear transformations, but you don't have to do that here.
     \begin{enumerate}
      \item Prove that $f$ is one-to-one, but not onto.
      \item Prove that $g$ is onto, but not one-to-one.
      \item What can you say about $f\circ g$ and $g\circ f$?
      \medskip
    
      {\sc Note:} Contrast the situation in this problem with that in problem 2. 
     \end{enumerate}
  \end{prob}
\end{document}
