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\begin{document}
\begin{center}
{\bf Homework 8}\\
(due Wednesday, April 4)
\end{center}
\vspace*{.3in}

\noindent 

\begin{prob}
Let $V$ be a vector space over some field $\F$ and $H\subset V$ be a subspace.
\begin{enumerate}
\item Show that scalar multiplication $\lambda\cdot\tilde\vv := \widetilde{\lambda\vv}$ is well-defined on $V/H$.
\item Suppose that $V$ is finite dimensional and $\cB = \cB_H\cup \cB'$ is a basis for $V$ obtained by extending a basis $\cB_H$ for $H$.  Show that the quotient map $\pi$ is injective on $\cB'$ and that $\pi(\cB')$ is a basis for $V/H$.
\item
Show that $V$ is isomorphic to $V/H\times H$.  Is the isomorphism `canonical'?  Explain.
\end{enumerate}
\end{prob}

\begin{prob} (Inner products don't cooperate with quotients)
Consider $V=\R^2$ with the standard inner (dot) product, and set $H=\spn\{\ve_1\}\subset V$.  One might try to define an inner product on $V/H$ by declaring that
$$
\pair{\tilde\vv}{\tilde\vw} := \pair{\vv}{\vw}
$$
Show by example that this won't work.  
\end{prob}


\begin{prob}
Let $m\geq 2$ be an integer.  Recall that $a,b\in\Z$ are \emph{congruent modulo $m$} ($a\equiv b\mod m$) if $b-a$ is an integer multiple of $m$.  We saw in class that this is an equivalence relation on $\Z$.  The quotient set (i.e. set of equivalence classes) for congruence modulo $m$ is commonly denoted $\Z/m\Z$. 
\begin{enumerate}
 \item Describe (with justification) the elements of $\Z/m\Z$.
 \item Show that addition and multiplication are well-defined on $\Z/m\Z$ (e.g. if we let $\tilde a,\tilde b$ denote the equivalence classes of $a,b\in\Z$ then it is unambiguous to declare that $\tilde a + \tilde b := \widetilde{a+b}$).
\item Show by example that exponentiation $a^b$ is \emph{not} well-defined on $\Z/m\Z$.
\end{enumerate}
\end{prob}

\begin{prob}
The set 
$$
\cB = \{\vv_1,\dots,\vv_4\} := \left\{\cvec{1 \\ 0 \\ 1 \\ -1}, \cvec{0\\ 1\\ 0\\ 1}, \cvec{-1\\ 1\\ 0\\ 0}, \cvec{1\\ 1\\ 0\\ 1}\right\}
$$
is a basis for $\R^4$.  Verify that the subspace $H = \spn\{\vv_1,\vv_2\}$ is invariant under the linear operator $T:\R^4\to\R^4$ given by $T(\vx) = A\vx$, where
$$
A = \left[\begin{matrix}
0 & -2 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 4 & -6 & -6 \\ -3 & -8 & 15 & 11
          \end{matrix}\right].
$$
Then find the matrices for $T|_H$ and $\tilde T:V/H\to V/H$ relative to the bases $\{\vv_1,\vv_2\}$ and $\{\tilde\vv_3,\tilde\vv_4\}$, respectively.  You'll probably want to use a calculator or computer to help out with the computations here.
\end{prob}

\begin{prob}
Suppose that $T:V\to V$ is a linear operator with distinct eigenvalues $\lambda,\lambda' \in \F$.  Let $H_\lambda$ and $H_{\lambda'}$ denote the corresponding eigenspaces, and let $\tilde V = V/H_{\lambda}$.  Show that $T:V\to V$ induces a well-defined linear operator $\tilde T:\tilde V\to \tilde V$ and that $\lambda'$ is an eigenvalue of $\tilde T$ whose eigenspace has dimension equal to that of $H_{\lambda'}$. (Hint: let $\rho:V\to \tilde V$ be the quotient map.  Show that $\rho$ sends $H_{\lambda'}$ isomorphically onto $\tilde H_{\lambda'}$.)
\end{prob}


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