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\begin{document}
\centerline{Math 201 Homework for Tuesday, Week 7}
\bigskip
Due: Tuesday, October 12.

\begin{prob}
  Recall that if~$P,Q\in M_{m\times n}(F)$, then their sum $P+Q\in M_{m\times
  n}(F)$
  is defined to be the matrix with $i,j$-th entry
  \[
    (P+Q)_{ij}:=P_{ij}+Q_{ij},
  \]
  On the other hand, if $P\in M_{m\times \ell}(F)$ and~$Q\in M_{\ell\times
  n}(F)$ (note the change in dimensions), then their
  product~$PQ\in M_{m\times n}(F)$ is defined to be the matrix
  whose $i,j$-th entry is
  \[
    (PQ)_{ij}:=\sum_{k=1}^{\ell} P_{ik}Q_{kj}.
  \]

  Let $A\in M_{m\times \ell}(F)$, and let $B,C\in M_{\ell\times n}(F)$. Using only the
  definition of matrix addition and matrix multiplication, prove that
  \[
    A(B+C)=AB+AC.
  \]
  You do this by proving that the $i,j$-th entries on both sides are equal.
  \textbf{Please use summation notation, and be careful to specify the correct
  starting and ending points for the summation.} (The result you are proving is called the {\em left
distributivity} property of matrix multiplication.)
\end{prob}

\begin{prob}  For the following, recall that just as an example usually does not
  constitute a proof that something is true, a general discussion does not
  usually suffice to proof that something is not true. In the latter case, it is
  fine (but not necessary) to give a general discussion, but in the end, you
  should provide a concrete and simple counterexample.
  \begin{enumerate}[(a)]
    \item Prove that matrix multiplication of $2\times 2$ matrices does not satisfy the commutative law, $AB=BA$.
    \item Prove that matrix multiplication of $2\times 2$ matrices does not satisfy left cancellation.\\
      \emph{Cancellation:} If $AB=AC$ and $A$ is not the zero matrix, then $B=C$.
  \end{enumerate}
\end{prob}

\begin{prob}
  Let $\mathcal{P}_n(\R)$ be the vector space of polynomials in $x$ of degree at
  most $n$ with coefficients in $\R$. Let $f\colon \mathcal{P}_2(\R) \to
  \mathcal{P}_2(\R)$ and $g\colon \mathcal{P}_2(\R) \to \R^3$ be the linear
  transformations respectively defined as 
  \[
    f(p(x))=(3+x)p'(x)+2p(x) \quad \text{and} \quad g(a+bx+cx^2)=(a+b,c,a-b).
  \]
  Let $\mathcal{B}=\langle 1,x,x^2\rangle$ and $\mathcal{D}$ be the standard ordered basis for $\R^3$.
  \begin{enumerate}[(a)]
    \item Compute the matrix representing $f$ with respect to the basis $\mathcal{B}$ for both the domain and codomain.
    \item Is $f$ one-to-one? Is it onto?
    \item Compute the matrix representing $g$ with respect to the bases $\mathcal{B}$ and~$\mathcal{D}$.
    \item Compute the matrix representing $g\circ f$ with respect to the bases
      $\mathcal{B}$ and~$\mathcal{D}$. Then use Theorem 2.7 (Chapter Three,
      Section IV)  to verify your result.  (This theorem  says the composition
      of linear maps is represented by the matrix product of the representatives
      of the linear maps.)
  \end{enumerate}
\end{prob}
\end{document}

