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\begin{document}
\centerline{Math 201 Homework for Friday, Week 5}
\bigskip

\textbf{Due:} Friday, October 1.

\begin{prob}
  For the following functions $f$:
  \begin{enumerate}[(i)]
    \item prove that $f$ is a linear transformation,
    \item find bases for $\mathcal{N}(f)$ and $\mathcal{R}(f)$, and
    \item compute the nullity and the rank of $f$.
  \end{enumerate}
  \begin{enumerate}
    \item $f\colon \R^3 \to \R^2$ defined by $f(x,y,z)=(x-y+z,2y+z)$. 
    \item $f\colon \mathcal{P}_2(\R) \to \mathcal{P}_3(\R)$ defined by $f(p(x))=x\cdot p(x)+2p'(x)$.

      (Recall that $\mathcal{P}_n(F)$ denotes the vector space of polynomials with coefficients in $F$ of degree less or equal to $n$. Here, $p'(x)$ denotes the standard derivative from calculus.)
  \end{enumerate}
\end{prob}

\begin{prob}
  Let $V$, $W$ and $U$ be finite-dimensional vector spaces over $F$, and let $f\colon V \to W$, $g\colon W \to U$ be a linear transformations. One can prove\footnote{You don't have to turn that in, but it is a good exercise for you to try.} that $g\circ f$ is also a linear transformation. 
  \begin{enumerate}
    \item Show that 
      \[\rk(g\circ f) \leq \min\{\rk(f),\rk(g)\}.\]
    \item Give an example in which the inequality is strict.
  \end{enumerate}
\end{prob}
    
\begin{prob}
  Let $V$ and $W$ be vector spaces over $F$, and let $f\colon V \to W$ be linear an isomorphism (bijective linear transformation). Let $g\colon W \to V$ be the inverse function to $f$, that is, $g$ satisfies that $g\circ f= \id_V$ and $f\circ g=\id_W$. Prove that $g$ is a linear transformation.
\end{prob}
\end{document}


