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\begin{document}
\begin{center}
{\bf Homework 5}\\
(due Wednesday, September 28)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:} 
\begin{description}
\item[1.3] 3.3d (Suppose that $n=3$ in this problem)
\item[1.5] 5.1, 5.3, 5.5 
\item[2.4] 4.1 (please show your work here)
\end{description}

\begin{prob}
Let $A,B,C$ be sets and $f:A\to B, g:B\to C$ be functions.  
Show that
\begin{enumerate}
\item if $f$ and $g$ are injective, then so is $g\circ f$.
\item if $f$ and $g$ are surjective, then so is $g\circ f$.
\end{enumerate} 
\end{prob}

\begin{prob}
Let $V$ be a real vector space and $\vv_1,\dots,\vv_n$ be vectors.  Define a linear transformation $\phi:\R^n\to V$ by 
$$
\phi(x_1,\dots,x_n) = x_1\vv_1 + \dots + x_n\vv_n.
$$
Under what condition is $\varphi$ injective? surjective?  Justify your answers!
\end{prob}

\begin{prob}
Describe a vector space isomorphism $T:V\to W$ between each of the following 
pairs $V,W$ of real vector spaces.  You needn't justify that what you describe
is linear (as long as it is!), and to justify that it is bijective, it will
suffice if you just describe the inverse $T^{-1}:W\to V$. 
\begin{enumerate}
\item $V=\C$ and $W = P_1(\R)$ (the space of polynomials of degree at 
most $1$).
\item $V = P_1(\R)$ and $W$ is the subspace of $P_2(\R)$
consisting of polynomials with $1$ as a root.
\item the vector space of all continuous functions $f:[0,1]\to\R$ to the 
vector space of all continuous functions $g:[-2,2]\to\R$.
(\emph{Notational remark: it's often a little awkward to describe a linear
  transformation that turns one function into another function.  I find things 
  like ``let $(T(f))(x) = f''(\sin x) - 3x^2$'' a little confusing.  To me
  it seems easier to say instead ``let $T(f) = g$ where $g(x) = 
  f''(\sin x) - 3x^2$.''})
\end{enumerate}
\end{prob}

\begin{prob}
Let $T:V\to W$ be a linear transformation, and $A\subset V$, $B\subset W$
be subsets of the domain and codomain, respectively.
\begin{enumerate}
\item Show that if $T$ is injective and $T(A)$ is a subspace, then $A$ must be
  a subspace.
\item Show that if $T$ is surjective and $T^{-1}(B)$ is a subspace, then $B$
  must be a subspace.
\item Give an example illustrating that the assertion in the first part can fail if $T$ is not injective.  Likewise, give an example illustrating that the assertion in the second part can fail if we drop the surjectivity assumption. 
\end{enumerate}
\end{prob}


\begin{prob}
(Complex numbers as $2\times 2$ matrices)
Recall the following facts about $\C$ from lecture and previous homework
(i.e. you don't have to prove them again):
\begin{itemize}
\item As a vector space over $\R$, we have $\dim \C = 2$ and, more
  specifically, $\cB=\{1,i\}$ is a basis for $\C$;
\end{itemize}
and if $T:\C\to\C$ is complex linear then
\begin{itemize}
\item $T$ is given by $T(z) = az$ for some $a\in\C$ and all $z\in\C$.
\item $T$ is also real linear.
\end{itemize}
Now do each of the following concerning $T$.
\begin{enumerate}
\item find the matrix $A=[T]_{\cB\cB}\in M_{2\times 2}(\R)$ for
$T$ relative to the basis $\cB$. That is,
give the entries of $A$ in terms of the real and imaginary parts of $a$.
\item Let $B\in M_{2\times 2}(\R)$ be the matrix corresponding to multiplication by
another complex number $b\in\C$.  Show by direct computation that $AB$ is the matrix
for multiplication by $ab$.  Why is this to be expected?
\item What unusual fact does this imply about $BA$? (no further computation allowed! At least,
don't show it to us)
\end{enumerate}
\end{prob}

\end{document}
\end

