Commit 2f5303c0 authored by Jim Hefferon's avatar Jim Hefferon

homogeom

parent 65bd3df9
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......@@ -23,7 +23,7 @@ pref,%
%dummy_chapter,% for printing the answers to Topic exercises
gr1,gr2,gr3,cas,leontif,ppivot,network,%
vs1,vs2,vs3,fields,crystal,voting,dimen,%
map1,map2,map3,map4,map5,map6,lstsqs,erlang,markov,homogeom,%
map1,map2,map3,map4,map5,map6,lstsqs,homogeom,markov,erlang,%
det1,det2,det3,cramer,detspeed,projplane,%
jc1,jc2,jc3,jc4,powers,pops,recur,%eigengeom,prinaxis,%
appen,%
......
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# bridges.sage
# data on bridges and tolls
# http://costoftolls.com/Tolls_in_New_York.html
# http://maps.google.com/
# 2012-Jan-07 JH
# Bring up sage in a terminal, then call
# load "bridges.sage"
#
# Holland Tunnel 12.00 one way, 7
# Lincoln Tunnel 12.00 one way, 2
# George Washington Bridge 12.00 one way, 8
# Bear Mountain Bridge 1.00, 47
# Mid Hudson Bridge, Highland, NY, 1.00, 82
# Gov. Thomas E. Dewey Thruway bridge over hudson toll part of larger road toll
# Kingston-Rhinecliff Bridge, 1.00 , 102
# Newburgh-Beacon Bridge, 1.00, 67
# Rip Van Winkle Bridge, 1.00, 120
# Tappan Zee Bridge, 5.00 one way, 27
# Verrazano-Narrows Bridge, 13.00 one way, 16
data=[
[2,6], # Lincoln Tunnel
[7,6], # Holland Tunnel
[8,6], # George Washington Bridge
[16,6.5], # Verrazano-Narrows Bridge
[27,2.5], # Tappan Zee Bridge
[47,1], # Bear Mountain Bridge
[67,1], # Newburgh-Beacon Bridge
[82,1], # Mid Hudson Bridge
[102,1], # Kingston-Rhinecliff Bridge
[120,1], # Rip Van Winkle Bridge
]
var('slope,intercept')
model(x) = slope*x+intercept
# use this to find the slope and intercept: find_fit(data,model)
g=points(data)+plot(model(intercept=find_fit(data,model)[0].rhs(),slope=find_fit(data,model)[1].rhs()),(x,0,140),color='red',figsize=3)
g.save("bridges.png")
\ No newline at end of file
......@@ -2435,7 +2435,7 @@ beginfig(50) % rotation ccw
numeric w; %horizontal scaling factor
u:=.15in; w:=u; v:=u;
save codomain_shift; pair codomain_shift; codomain_shift=(12.5w,0v);
save codomain_shift; pair codomain_shift; codomain_shift=(15.5w,0v);
% the axes
save xmin, xmax, ymin, ymax; numeric xmin, xmax, ymin, ymax;
xmin = -.5w; xmax = 3.5w; ymin = -.5v; ymax = 3.5v;
......@@ -2449,7 +2449,7 @@ beginfig(50) % rotation ccw
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2531,7 +2531,7 @@ beginfig(51) % projection to xz-axis
%z30 = .25*(z10+z11+z12+z13)+.5*codomain_shift;
z30 = (point .5 of zaxis) shifted (.5*codomain_shift);
label(btex {\tiny $\mapsunder{\colvec{x \\ y \\ z} \mapsto \colvec{x \\ 0 \\ z}}$} etex,z30);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y \\ z} \mapsto \colvec{x \\ 0 \\ z}}$ etex,z30);
% clean up the observer
free_vect(lpn);
......@@ -2566,7 +2566,7 @@ beginfig(52) % tripling the x's
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{3x \\ y}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{3x \\ y}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2605,7 +2605,7 @@ beginfig(53) % negative doubling the x's
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{-2x \\ y}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{-2x \\ y}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2644,7 +2644,7 @@ beginfig(54) % negative doubling the x's
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{y \\ x}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{y \\ x}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2688,7 +2688,7 @@ beginfig(55) % skew's action
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{x \\ 2x+y}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{x \\ 2x+y}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2735,7 +2735,7 @@ beginfig(56) % skew on unit square
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{x \\ 2x+y}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{x \\ 2x+y}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......@@ -2783,7 +2783,7 @@ beginfig(57) % second skew on unit square
xpart(point .5 of (xaxis shifted codomain_shift))];
y1 = .5[ypart(point .5 of yaxis),
ypart(point .5 of (yaxis shifted codomain_shift))];
label(btex {\tiny $\mapsunder{\colvec{x \\ y} \mapsto \colvec{x+2y \\ y}}$} etex,z1);
label(btex $\xrightarrow{\scriptsize \colvec{x \\ y} \mapsto \colvec{x+2y \\ y}}$ etex,z1);
% the vectors
z0 = (0w,0v);
......
......@@ -76,7 +76,7 @@ The obvious example is this \emph{translation}.\index{translation}
\begin{equation*}
\colvec{x \\ y}
\quad\mapsto\quad
\colvec{x \\ y}+\colvec{1 \\ 0}=\colvec{x+1 \\ y}
\colvec{x \\ y}+\colvec[r]{1 \\ 0}=\colvec{x+1 \\ y}
\end{equation*}
However,
this example turns out to be the only example, in the
......@@ -211,20 +211,20 @@ from $\vec{e}_1$
\begin{tabular}{@{}c@{}}\includegraphics{ch3.62}\end{tabular}
\qquad
$\rep{t}{\stdbasis_2,\stdbasis_2}=
\begin{pmatrix}
\begin{mat}
a &-b \\
b &a
\end{pmatrix}$
\end{mat}$
\end{center}
and one where is is mapped a quarter circle counterclockwise.
\begin{center}
\begin{tabular}{@{}c@{}}\includegraphics{ch3.63}\end{tabular}
\qquad
$\rep{t}{\stdbasis_2,\stdbasis_2}=
\begin{pmatrix}
\begin{mat}
a &b \\
b &-a
\end{pmatrix}$
\end{mat}$
\end{center}
We can geometrically describe these two cases.
......@@ -327,18 +327,18 @@ More on Klein and the Erlanger Program is in \cite{Yaglom}.
\item
Decide if each of these is an orthonormal matrix.
\begin{exparts}
\partsitem $\begin{pmatrix}
\partsitem $\begin{mat}[r]
1/\sqrt{2} &-1/\sqrt{2} \\
-1/\sqrt{2} &-1/\sqrt{2}
\end{pmatrix}$
\partsitem $\begin{pmatrix}
\end{mat}$
\partsitem $\begin{mat}[r]
1/\sqrt{3} &-1/\sqrt{3} \\
-1/\sqrt{3} &-1/\sqrt{3}
\end{pmatrix}$
\partsitem $\begin{pmatrix}
\end{mat}$
\partsitem $\begin{mat}[r]
1/\sqrt{3} &-\sqrt{2}/\sqrt{3} \\
-\sqrt{2}/\sqrt{3} &-1/\sqrt{3}
\end{pmatrix}$
\end{mat}$
\end{exparts}
\begin{answer}
\begin{exparts}
......@@ -363,38 +363,38 @@ More on Klein and the Erlanger Program is in \cite{Yaglom}.
\partsitem
$\colvec{x \\ y}
\mapsto
\begin{pmatrix}
\begin{mat}
x\cdot\cos(\pi/6)-y\cdot\sin(\pi/6) \\
x\cdot\sin(\pi/6)+y\cdot\cos(\pi/6)
\end{pmatrix}
+\colvec{0 \\ 1}
=\begin{pmatrix}
\end{mat}
+\colvec[r]{0 \\ 1}
=\begin{mat}
x\cdot(\sqrt{3}/2)-y\cdot(1/2)+0 \\
x\cdot(1/2)+y\cdot\cos(\sqrt{3}/2)+1
\end{pmatrix}$
\end{mat}$
\partsitem The line $y=2x$ makes an angle of $\arctan(2/1)$
with the $x$-axis.
Thus $\sin\theta=2/\sqrt{5}$ and $\cos\theta=1/\sqrt{5}$.
\begin{equation*}
\colvec{x \\ y}
\mapsto
\begin{pmatrix}
\begin{mat}
x\cdot(1/\sqrt{5})-y\cdot(2/\sqrt{5}) \\
x\cdot(2/\sqrt{5})+y\cdot(1/\sqrt{5})
\end{pmatrix}
\end{mat}
\end{equation*}
\partsitem
$\colvec{x \\ y}
\mapsto
\begin{pmatrix}
\begin{mat}
x\cdot(1/\sqrt{5})-y\cdot(-2/\sqrt{5}) \\
x\cdot(-2/\sqrt{5})+y\cdot(1/\sqrt{5})
\end{pmatrix}
+\colvec{1 \\ 1}
=\begin{pmatrix}
\end{mat}
+\colvec[r]{1 \\ 1}
=\begin{mat}
x/\sqrt{5}+2y/\sqrt{5}+1 \\
-2x/\sqrt{5}+y/\sqrt{5}+1
\end{pmatrix}$
\end{mat}$
\end{exparts}
\end{answer}
\item \label{exer:IsometryFacts}
......@@ -437,18 +437,18 @@ More on Klein and the Erlanger Program is in \cite{Yaglom}.
Check that these two computations yield the same
first two components.
\begin{equation*}
\begin{pmatrix}
\begin{mat}
a &c \\
b &d
\end{pmatrix}
\end{mat}
\colvec{x \\ y}
+\colvec{e \\ f}
\qquad
\begin{pmatrix}
\begin{mat}
a &c &e \\
b &d &f \\
0 &0 &1
\end{pmatrix}
\end{mat}
\colvec{x \\ y \\ 1}
\end{equation*}
(These are
......
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......@@ -3261,10 +3261,10 @@ Note, as a check, that this result is indeed in $P$.
See the third item of \nearbyexercise{exer:AlgOfPerps}.)
\partsitem Generalize that to apply to any $\map{f}{\Re^n}{\Re^m}$.
\end{exparts}
This, and related results, is the
In \cite{Strang93}
this is called the
\definend{Fundamental Theorem of Linear Algebra}%
\index{Fundamental Theorem!of Linear Algebra}
in \cite{Strang93}.
\begin{answer}
\begin{exparts}
\partsitem The representation of
......
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