Commit 148e8759 authored by Jim Hefferon's avatar Jim Hefferon

concrete style

parent b7e99acf
......@@ -6,10 +6,9 @@
verbatimtex
%&latex
\documentclass{book}
\usepackage{dvidrv} \usepackage{hrefout}
\usepackage{bookjh,linalgjh}
\usepackage{color}
\usepackage{verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
This diff is collapsed.
......@@ -9,10 +9,11 @@
\usepackage[no]{hrefout} % allow selection of hyperref from command line
\fi
\usepackage{verbatim}
\usepackage{xr}
\usepackage[single,write]{bookans}
\usepackage{bookjh} %,linalgjh}
\usepackage{verbatim}
\usepackage{bookjhconcrete}
% \usepackage{bookjh}
%\usepackage{makeidx}
% \usepackage{showidx} \typeout{REMOVE showidx on final runs!}
\includeonly{
......@@ -28,7 +29,9 @@ bib%
%test%
}%
\usepackage{makeidx}\makeindex
\overfullrule=5pt % if a draft
% Uncomment this to show overfull boxes with a black box in the margin
% \overfullrule=5pt % if a draft
\begin{document}
\pagenumbering{roman}
......
......@@ -392,10 +392,10 @@
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
ka_{i,1}x_1 &+ &ka_{i,2}x_2 &+ &\cdots &+ &ka_{i,n}x_n
&= &kd_i \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{equation*}
......@@ -427,9 +427,9 @@
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &+ &a_{i,n}x_n &= &d_i \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m
\end{linsys}
......@@ -441,12 +441,12 @@
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1\hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &\cdots &+ &a_{i,n}x_n &= &d_i\hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
(ka_{i,1}+a_{j,1})x_1 &+ &\cdots &+ &(ka_{i,n}+a_{j,n})x_n
&= &kd_i+d_j \hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m\hfill\hbox{}
\end{linsys}
......@@ -489,11 +489,11 @@
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &\cdots &+ &a_{i,n}x_n &= &d_i \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{j,1}x_1 &+ &\cdots &+ &a_{j,n}x_n &= &d_j \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m\hfill\hbox{}
\end{linsys}
......@@ -526,14 +526,14 @@
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{\rho_i\leftrightarrow\rho_j}\;
\grstep{\rho_j\leftrightarrow\rho_i}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......@@ -542,14 +542,14 @@
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{k\rho_i}\;
\grstep{(1/k)\rho_i}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......@@ -558,14 +558,14 @@
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{k\rho_i+\rho_j}\;
\grstep{-k\rho_i+\rho_j}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......@@ -484,7 +484,7 @@
\par
\vskip 0\p@ \fi %was 20\p@
\interlinepenalty\@M
\Huge \bfseries \textcolor{black}{#1}\par\nobreak
\Huge \bfseries\textcolor{black}{#1}\par\nobreak
\vskip 50\p@ % was 30\p@
}}
......@@ -792,5 +792,4 @@
\end{tabular}}}
}
\endinput
This diff is collapsed.
......@@ -3,15 +3,9 @@
verbatimtex
%&latex
\documentclass{book}
\typeout{PAST documentclass{book}}
\usepackage{dvidrv}
\typeout{PAST dvidrv}
\usepackage{hrefout}
\typeout{PAST hrefout}
\usepackage{bookjh,linalgjh}
\typeout{PAST bookjh}
\usepackage{verbatim}
\typeout{PAST verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
......@@ -3,9 +3,10 @@
verbatimtex
%&latex
\documentclass{book}
\usepackage{dvidrv} \usepackage{hrefout}
\usepackage{bookjh,linalgjh}
\usepackage{verbatim}
\documentclass{book}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
......@@ -3,9 +3,9 @@
verbatimtex
%&latex
\documentclass{book}
\usepackage{dvidrv} \usepackage{hrefout}
\usepackage{bookjh,linalgjh}
\usepackage{verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
......@@ -3,9 +3,9 @@
verbatimtex
%&latex
\documentclass{book}
\usepackage{dvidrv} \usepackage{hrefout}
\usepackage{bookjh,linalgjh}
\usepackage{verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
......@@ -6,10 +6,9 @@
verbatimtex
%&latex
\documentclass{book}
\usepackage{dvidrv} \usepackage{hrefout}
\usepackage{bookjh,linalgjh}
\usepackage{color}
\usepackage{verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
% conc.sty
% Concrete fonts for Linear Algebra by Jim Hefferon
\usepackage[T1]{fontenc}
\usepackage[boldsans]{ccfonts}
\usepackage[euler-digits]{eulervm}
\linespread{1.04} % approximately
% \usepackage{mathdots}
% for typewriter text from http://tex.stackexchange.com/questions/18715/suggestions-for-typewriter-font-to-match-concrete-and-euler
\usepackage[scaled=0.88]{luximono}
......@@ -2,8 +2,9 @@
% http://joshua.smcvt.edu/linalg.html
% 2001-Jun-12
\section{Laplace's Expansion}
\textit{(This section is optional.
Later sections do not depend on this material.)}
% \textit{This section is optional.
% Later sections do not depend on this material.}
Determinants are a font of interesting and amusing formulas.
Here is one that is often used
......@@ -209,7 +210,7 @@ these are the \( 1,2 \) and \( 2,2 \) cofactors.
\index{determinant!Laplace expansion}%
\index{Laplace expansion!computes determinant}
Where \( T \) is an \( \nbyn{n} \) matrix, we can find
the determinant by expanding by cofactors on either
the determinant by expanding by cofactors on any
row~$i$ or column~$j$.
\begin{align*}
\deter{T}
......
......@@ -98,7 +98,7 @@ A \definend{system of linear equations}\index{linear equation!system of}%
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
a_{2,1}x_1 &+ &a_{2,2}x_2 &+ &\cdots &+ &a_{2,n}x_n &= &d_2 \\
& & & & & & &\vdots \\
& & & & & & &\vdotswithin{=} \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{equation*}
......@@ -239,28 +239,28 @@ The other two
cases are \nearbyexercise{ex:ProveGaussMethod}.
Consider the swap of row~$i$ with row~$j$.
{\renewcommand{\arraystretch}{.75}
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\
& & & & & & &\vdots \\
a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\
& & & & & & &\vdots \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m
\end{linsys}
\grstep{}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\
& & & & & & &\vdots \\
a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\
& & & & & & &\vdots \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m
\end{linsys}
\end{equation*} }% end change of arraystretch
% {\renewcommand{\arraystretch}{.75}
% \begin{equation*}
% \begin{linsys}{4}
% a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m
% \end{linsys}
% \grstep{}
% \begin{linsys}{4}
% a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\
% & & & & & & &\shortvdotswithin{=} \\
% a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m
% \end{linsys}
% \end{equation*} }% end change of arraystretch
The tuple \( (s_1,\ldots\,,s_n) \)
satisfies the system before the swap
if and only if substituting the values for the
......@@ -1270,10 +1270,10 @@ a no response by showing that no solution exists.}
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
ka_{i,1}x_1 &+ &ka_{i,2}x_2 &+ &\cdots &+ &ka_{i,n}x_n
&= &kd_i \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{equation*}
......@@ -1305,9 +1305,9 @@ a no response by showing that no solution exists.}
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &+ &a_{i,n}x_n &= &d_i \\
& & & & & & &\vdots \\
& & & & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m
\end{linsys}
......@@ -1319,12 +1319,12 @@ a no response by showing that no solution exists.}
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1\hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &\cdots &+ &a_{i,n}x_n &= &d_i\hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
(ka_{i,1}+a_{j,1})x_1 &+ &\cdots &+ &(ka_{i,n}+a_{j,n})x_n
&= &kd_i+d_j \hfill \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m\hfill\hbox{}
\end{linsys}
......@@ -1367,11 +1367,11 @@ a no response by showing that no solution exists.}
\begin{equation*}
\begin{linsys}{4}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{i,1}x_1 &+ &\cdots &+ &a_{i,n}x_n &= &d_i \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{j,1}x_1 &+ &\cdots &+ &a_{j,n}x_n &= &d_j \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &=
&d_m\hfill\hbox{}
\end{linsys}
......@@ -1414,14 +1414,14 @@ a no response by showing that no solution exists.}
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{\rho_i\leftrightarrow\rho_j}\;
\grstep{\rho_j\leftrightarrow\rho_i}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......@@ -1430,14 +1430,14 @@ a no response by showing that no solution exists.}
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{k\rho_i}\;
\grstep{(1/k)\rho_i}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......@@ -1446,14 +1446,14 @@ a no response by showing that no solution exists.}
\begin{eqnarray*}
\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
&\grstep{k\rho_i+\rho_j}\;
\grstep{-k\rho_i+\rho_j}
&\begin{linsys}{3}
a_{1,1}x_1 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\
& & & & &\vdots \\
& & & & &\shortvdotswithin{=} \\
a_{m,1}x_1 &+ &\cdots &+ &a_{m,n}x_n &= &d_m
\end{linsys}
\end{eqnarray*}
......
......@@ -341,15 +341,14 @@ and parametrize \( x=2-y/2-z/2 \).
% +y\cdot\colvec[r]{-1/2 \\ 1 \\ 0}
% +z\cdot\colvec[r]{-1/2 \\ 0 \\ 1}\suchthat y,z\in\Re}
% \end{equation*}
Generalizing from lines and planes, we define a
\definend{\( k \)-dimensional linear surface}\index{linear surface}
(or \definend{\( k \)-flat}\index{flat})
in $\Re^n$ to be
Generalizing, a set of the form
$\set{\vec{p}+t_1\vec{v}_1+t_2\vec{v}_2+\cdots+t_k\vec{v}_k
\suchthat t_1,\ldots ,t_k\in\Re}$
where \( \vec{v}_1,\ldots,\vec{v}_k\in\Re^n \).
For example, in $\Re^4$,
where \( \vec{v}_1,\ldots,\vec{v}_k\in\Re^n \)
and $k\leq n$ is a
\definend{\( k \)-dimensional linear surface}\index{linear surface}
(or \definend{\( k \)-flat}\index{flat}).
For example, in $\Re^4$
\begin{equation*}
\set{\colvec[r]{2 \\ \pi \\ 3 \\ -0.5}
+t\colvec[r]{1 \\ 0 \\ 0 \\ 0}
......
......@@ -82,8 +82,7 @@ Delightful presentations that take this approach from the start are in
\subsectionoptional{Factoring and Complex Numbers; A Review}
\textit{
This subsection is a review only and we take the main results as known.
\textit{This subsection is a review only and we take the main results as known.
For proofs, see \cite{BirkhoffMaclane} or \cite{Ebbinghaus}.}
We consider a \definend{polynomial}\index{polynomial}
......
No preview for this file type
This diff is collapsed.
% pref.tex See http://joshua.smcvt.edu/linearalgebra
{\setlength{\parskip}{.7ex} % note the group-starting open curly
\bigskip
\vspace*{1.25in plus .2in minus .1in}
\noindent{\Huge\bf Preface}
\vspace*{.4in plus .1in minus .05in}
\par\noindent
% \bigskip
% \vspace*{1.25in plus .2in minus .1in}
% \noindent{\Huge\bf Preface}
% \vspace*{.4in plus .1in minus .05in}
% \par\noindent
\chapter*{Preface}
This book helps students to master the material of a standard
US undergraduate first course in Linear Algebra.
......
......@@ -3,8 +3,9 @@
verbatimtex
%&latex
\documentclass{book}
%\usepackage{bookjh,linalgjh}
\usepackage{verbatim}
\usepackage{bookjhconcrete} % uncomment if using bookjhconcrete.sty for book
% \usepackage{bookjh} % uncomment if using bookjh.sty for book
\usepackage{linalgjh}
\begin{document}
etex
......
......@@ -9,17 +9,17 @@ Gauss' method systematically takes linear combinations of the rows.
With that insight, we now move to a general study of linear
combinations.
We need a setting for this study.
At times in the first chapter, we've combined vectors from $\Re^2$,
We need a setting.
At times in the first chapter we've combined vectors from $\Re^2$,
at other times vectors from $\Re^3$,
and at other times vectors from even higher-dimensional spaces.
Thus, our first impulse might be
So our first impulse might be
to work in $\Re^n$, leaving $n$ unspecified.
This would have the advantage that any of the results
would hold for $\Re^2$ and for $\Re^3$ and for many other spaces,
simultaneously.
But, if having the results apply to many spaces at once is
But if having the results apply to many spaces at once is
advantageous then sticking only to $\Re^n$'s is overly restrictive.
We'd like the results to also apply to combinations of row vectors,
as in the final section of the first chapter.
......@@ -42,17 +42,16 @@ combinations \ldots'', will be stated as
Such a statement describes at once what
happens in many spaces.
The step up in abstraction from studying a single space at a time
to studying a class of spaces can be hard to make.
To understand its advantages, consider this analogy.
To understand the advantages of moving from studying a single space at a time
to studying a class of spaces, consider this analogy.
Imagine that the government made laws one person at a time:
``Leslie Jones can't jay walk.''
That would be a bad idea;
statements have the virtue of economy when they apply to many cases at once.
Or, suppose that they ruled, ``Kim Ke must stop when passing
the scene of an accident.''
Or suppose that they ruled, ``Kim Ke must stop when passing
an accident.''
Contrast that with, ``Any doctor must stop when passing
the scene of an accident.''
an accident.''
More general statements, in some ways, are clearer.
......
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