\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{amssymb}

%\usepackage[textheight=8.5in, margin=1.6in]{geometry}
\usepackage[total={6in,8in}]{geometry}

%\usepackage{tikz}
%\usepackage{pgfplots}
%\tikzstyle{ball} = [circle,shading=ball, ball color=black,
%    minimum size=1mm,inner sep=1.3pt]

%\usepackage{graphicx}

\usepackage{enumitem}
\setlist[enumerate]{itemsep=2pt,leftmargin=*,label={(\alph*)}}

%\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\ve}{\varepsilon}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Span}{\mathrm{Span}}
\DeclareMathOperator{\rk}{\mathrm{rank}}
\DeclareMathOperator{\trace}{\mathrm{tr}}
\DeclareMathOperator{\im}{\mathrm{im}}
\DeclareMathOperator{\Sym}{\mathrm{Sym}}
\DeclareMathOperator{\Hom}{\mathrm{Hom}}

\newcounter{problem}
\renewcommand \thesection {\theproblem}
\newenvironment{prob}[1][]
{
\refstepcounter{problem}\par\medskip
\def\myArg{#1}
  \ifx\myArg\empty
    \noindent \textsc{Problem~\theproblem.} \rmfamily
  \else
    \noindent \textsc{Problem~\theproblem\ (\textnormal{#1}).} \rmfamily
  \fi
}
{
\medskip

}

\parskip = 0.05in
\parindent = 0.0in

\pagestyle{empty}

\begin{document}

\centerline{Math 201 Homework for Tuesday, Week 2}
\bigskip

\textbf{Due:} upload your work to Gradescope by 3 p.m.~on Tuesday, September 7.

\begin{prob}
  Let~$L$ be the line in~$\R^3$ passing through the points~$(1,1,1)$
    and~$(2,7,4)$. 
    \begin{enumerate}
      \item Find a system of two linear equations whose solution set
	is~$L$.  Show your work. (\emph{Hint}: This will likely involve solving a system of linear equations.)
      \item Give a parametrization of~$L$ (you should only need one parameter).
    \end{enumerate}
\end{prob}

\begin{prob}
  Let~$H$ be the plane in ~$\R^3$ containing the points~$(1,1,0)$, $(1,5,-3)$,
    and~$(1,-2,4)$.  
    \begin{enumerate}
      \item Find a linear equation whose solution set is~$H$. Show your work.
      \item Give a parametrization of~$H$.
      \item What happens if we instead consider the three points $(1,1,0)$,
	$(1,5,-3)$ and $(1,-3,3)$? Is there such a plane? How does the process
	go differently for (a) and (b)?
    \end{enumerate}
\end{prob}

\begin{prob}
  Let~$H$ be the subset of vectors ~$(x_1,\dots,x_n)$ in ~$\R^n$ given by the
  set of solutions to the equation \[a_1x_1+\dots+a_nx_n=d,\] for some constants
  $a_1,\dots,a_n,d$, with at least one $a_i$ not equal to 0. Such set is called
  a \emph{hyperplane}.
    \begin{enumerate}
      \item Prove that this set of solutions can be parametrized using $n-1$ parameters. 
      \item What do you expect the dimension of $H$ to be? (We haven't defined dimension precisely, so use your intuition.) 
      \item How many points do you expect need to be given to determine a hyperplane in $\R^n$?
    \end{enumerate}
\end{prob}

\end{document}
