\documentclass[minion]{homework}
\newcommand{\Reals}{\mathbb{R}}

\doclabel{Math F651: Homework 8}
%\docauthor{Your name here.}
\docdate{Due: March 28, 2018}
\begin{document}
\begin{aproblems}


\hproblem 5.15

\hproblem 5.16 (Read Def. 5.45 first)

\hproblem 5.17

\hproblem 5.18

\hproblem 5.19 

If you have not seen the definition of quotient of vector space by a subspace, it's not a hard concept.  Given a vector space $X$ and a subspace $W$ we say
$x\sim z$ if $x-z\in W$, or alternatively if $x=z+w$ for some $w\in W$.  It's
easy to see that this is an equivalence relation and we write the equivalence
class of $x$ as $x+W$ rather than $[x]$.  We define $(x+W)+(z+W)$ by $(x+z)+W$
and $\alpha(x+W)=\alpha x+ W$.  You should show that these operations are well defined as a preamble to starting 5.19.  With these operations, the set of equivalence classes, written $X/W$, is a vector space, though you do not need to prove this.

\hproblem 5.22 (We sketched this in class; write the details down.)

\end{aproblems}
\end{document}

