Commit 472763bc authored by Jim Hefferon's avatar Jim Hefferon

three_ii

parent e793a090
......@@ -22,7 +22,7 @@ numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v;
numeric dot_width; dot_width = 1.2pt;
numeric dot_width; dot_width = 1.2pt; % size of a pt in a set
def place_P(expr loc) =
......
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......@@ -14,7 +14,9 @@ ahangle:=40;
input arrow
input jh3d
numeric dot_width; dot_width = 1.2pt; % size of a pt in a set
beginfig(1) % (2,2,1) and its projection
numeric u; %scaling factor
numeric v; %vertical scaling factor
......@@ -234,11 +236,17 @@ beginfig(4) % R^3 and R^2
save many_vecs; path many_vecs; many_vecs = (0,-4pt)--(0,4pt);
pickup pencircle scaled line_width_light;
drawarrow z0--z3 withcolor lightgray;
draw (many_vecs shifted z3) dashed withdots scaled 0.2;
pickup pencircle scaled dot_width;
draw (many_vecs shifted z3) dashed withdots scaled 0.35;
pickup pencircle scaled line_width_light;
drawarrow z0--z5 withcolor lightgray;
draw (many_vecs shifted z5) dashed withdots scaled 0.2;
pickup pencircle scaled dot_width;
draw (many_vecs shifted z5) dashed withdots scaled 0.35;
pickup pencircle scaled line_width_light;
drawarrow z0--z1 withcolor lightgray; % main vector
draw (many_vecs shifted z1) dashed withdots scaled 0.2;
pickup pencircle scaled dot_width;
draw (many_vecs shifted z1) dashed withdots scaled 0.35;
pickup pencircle scaled line_width_light;
% vectors in R^2 (shifted)
z20=z0 shifted rtwoshift;
......@@ -282,9 +290,10 @@ beginfig(5) % bean with mapsto
z0 = (1.4w,.8v); z1= (1.5w,.9v);
z2 = (.8w,2.1v); z3= (1w,2.1v);
z4 = (1.25w,3.3v); z5= (1.5w,3.1v);
pickup pencircle scaled line_width_light;
% pickup pencircle scaled line_width_light;
pickup pencircle scaled dot_width;
forsuffixes s = 0, 2, 4:
draw (many_pts shifted z[s]) dashed withdots scaled 0.3;
draw (many_pts shifted z[s]) dashed withdots scaled 0.4;
drawdot (z[s+1] shifted codshift);
draw_action_arrow((z[s]{(3,1)}..(z[s+1] shifted codshift)));
endfor
......@@ -445,15 +454,18 @@ beginfig(10) % bean picture showing nullspace
draw generic_set(2w,4.25v);
draw generic_set(2.1w,4.4v) shifted codshift;
save many_pts; path many_pts; many_pts = (0,-5pt)--(0,5pt);
save many_pts; path many_pts; many_pts = (0,-6pt)--(0,6pt);
z0 = (1.35w,2.1v); z1= (1w,2.1v);
pickup pencircle scaled line_width_light;
draw (many_pts shifted z0) dashed withdots scaled 0.3;
pickup pencircle scaled line_width_dark;
drawdot z0;
pickup pencircle scaled dot_width;
draw (many_pts shifted z0) dashed withdots scaled 0.4;
% pickup pencircle scaled line_width_dark;
pickup pencircle scaled dot_width;
% alresdy included as part of many_pts: drawdot z0;
label.lft(btex {\small $0_V$} etex,z0);
% pickup pencircle scaled dot_width;
drawdot (z1 shifted codshift);
label.rt(btex {\small $0_W$} etex,(z1 shifted codshift));
pickup pencircle scaled line_width_dark;
draw_action_arrow((z0{(3,1)}..(z1 shifted codshift)));
endfig;
......@@ -521,7 +533,8 @@ beginfig(11) % R^3 and R^2
for i=-3 step 1 until 3:
drawarrow z0--(z1+i*(0,.20v));
endfor
draw (many_vecs shifted (z1+(1pt,0))) dashed withdots scaled 0.25;
pickup pencircle scaled dot_width;
draw (many_vecs shifted (z1+(1pt,0))) dashed withdots scaled 0.35;
% vectors in R^2 (shifted)
z20=z0 shifted rtwoshift;
......@@ -555,7 +568,8 @@ beginfig(12) % `a' vectors mapped to a
pickup pencircle scaled line_width_light;
z3=whatever[z0,z1]; x3=-.5w;
z4=whatever[z0,z1]; y4=-.5v;
draw z3--z4 dashed withdots scaled 0.25;
pickup pencircle scaled dot_width;
draw z3--z4 dashed withdots scaled 0.35;
%label.rt(btex \makebox[0pt][l]{\small $x+y=a$} etex,z4);
z5=(.75w,1.25v)-(.5pt,.5pt);
pickup pencircle scaled line_width_light;
......@@ -568,9 +582,11 @@ beginfig(12) % `a' vectors mapped to a
pickup pensquare scaled line_width_light;
draw (xaxis shifted roneshift);
label.urt(btex {\small $\Re^1$} etex,(point infinity of xaxis) shifted roneshift);
pickup pencircle scaled 2line_width_dark;
% pickup pencircle scaled 2line_width_dark;
pickup pencircle scaled 2dot_width;
drawdot (z1 shifted roneshift) withcolor white;
pickup pencircle scaled line_width_light;
% pickup pencircle scaled line_width_light;
pickup pencircle scaled dot_width;
drawdot (z1 shifted roneshift);
label.bot(btex {\small $w$} etex,z1 shifted roneshift);
% The map
......
......@@ -276,7 +276,8 @@ They have the same action so they are the same function.
%</pf:HomoDetActOnBasis2>
\end{proof}
\begin{definition}
\begin{definition} \label{df:ExtendedLinearly}
%<*df:ExtendedLinearly>
Let $V$ and~$W$ be vector spaces and
let
$B=\sequence{\vec{\beta}_1,\ldots,\vec{\beta}_n}$
......@@ -289,6 +290,7 @@ $\vec{v}=c_1\vec{\beta}_1+\cdots+c_n\vec{\beta}_n$,
the action of the map is
$\hat{f}(\vec{v})=c_1\cdot f(\vec{\beta}_1)
+\cdots+c_n\cdot f(\vec{\beta}_n)$.
%</df:ExtendedLinearly>
\end{definition}
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-10; // fewer jaggies but very slow
settings.render=0; // for png: -10; // fewer jaggies but very slow
settings.maxtile=(10,10);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-10; // fewer jaggies but very slow
settings.render=0; // -10; // fewer jaggies but very slow
settings.maxtile=(10,10);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-10; // fewer jaggies but very slow
settings.render=0; // for png: -10; // fewer jaggies but very slow
settings.maxtile=(10,10);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3.5cm; size(0,height);
import settings;
settings.render=-3;
settings.render=0; // for png want -3
settings.maxtile=(20,20);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3.5cm; size(0,height);
import settings;
settings.render=-3;
settings.render=0; // for png want -3
settings.maxtile=(20,20);
import three;
......@@ -28,7 +28,9 @@ path plane_vec[];
for(int i=0; i < 3; ++i) {
plane_vec.push((0,0)--plane_endpt[i]);
}
pen color[]={rgb(0.9,0,0), rgb(0,0,0.75), rgb(0.8,0,0.75)};
pen color[]={rgb(0.9,0,0), rgb(0,0,0.75), rgb(0.8,0,0.75)};
real colorfactor = 0.2;
pen light_color[]={rgb(colorfactor*0.9,0,0), rgb(0,0,colorfactor*0.75), rgb(colorfactor*0.8,0,colorfactor*0.75)};
path space_vec[];
path vec1=(0,0)--(2,3);
path vec2=(0,0)--(-4,5);
......@@ -38,7 +40,7 @@ draw(path3(plane_endpt[0]--plane_endpt[2]),VECTORPEN+gray(0.9));
draw(path3(plane_endpt[1]--plane_endpt[2]),VECTORPEN+gray(0.9));
// draw and label the codomain plane vectors
for(int i=0; i < 3; ++i) {
draw(path3(plane_vec[i]),VECTORPEN+color[i]+gray(0.95),Arrow3(DefaultHead2,VECTORHEADSIZE,FillDraw));
draw(path3(plane_vec[i]),VECTORPEN+light_color[i]+gray(0.95),Arrow3(DefaultHead2,VECTORHEADSIZE,FillDraw));
}
// label(Label("{\tiny $\vec{w}_1$}"),path3(plane_vec[0]));
// label(Label("{\tiny $\vec{w}_2$}"),path3(plane_vec[1]),NW);
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3.5cm; size(0,height);
import settings;
settings.render=-3;
settings.render=0; // was -3
settings.maxtile=(20,20);
import three;
......@@ -29,6 +29,8 @@ for(int i=0; i < 3; ++i) {
plane_vec.push((0,0)--plane_endpt[i]);
}
pen color[]={rgb(0.9,0,0), rgb(0,0,0.75), rgb(0.8,0,0.75)};
real colorfactor = 0.2;
pen light_color[]={rgb(colorfactor*0.9,0,0), rgb(0,0,colorfactor*0.75), rgb(colorfactor*0.8,0,colorfactor*0.75)};
path space_vec[];
path vec1=(0,0)--(2,3);
path vec2=(0,0)--(-4,5);
......@@ -38,7 +40,7 @@ draw(path3(plane_endpt[0]--plane_endpt[2]),VECTORPEN+gray(0.9));
draw(path3(plane_endpt[1]--plane_endpt[2]),VECTORPEN+gray(0.9));
// draw and label the codomain plane vectors
for(int i=0; i < 3; ++i) {
draw(path3(plane_vec[i]),VECTORPEN+color[i]+gray(0.95),Arrow3(DefaultHead2,VECTORHEADSIZE,FillDraw));
draw(path3(plane_vec[i]),VECTORPEN+light_color[i]+gray(0.95),Arrow3(DefaultHead2,VECTORHEADSIZE,FillDraw));
}
// label(Label("{\tiny $\vec{w}_1$}"),path3(plane_vec[0]));
// label(Label("{\tiny $\vec{w}_2$}"),path3(plane_vec[1]),NW);
......@@ -52,9 +54,9 @@ path3 domain_vec[];
for (int i=0; i<3; ++i) {
for(int j=0; j < 5; ++j) {
inv_image_pt=(xpart(plane_endpt[i]), ypart(plane_endpt[i]), (j+starting_hgt[i])/6);
dot(inv_image_pt,inverse_image_pen);
dot(inv_image_pt,inverse_image_pen+gray(1-colorfactor));
inv_image_pt=(xpart(plane_endpt[i]), ypart(plane_endpt[i]), (-j+starting_hgt[i])/6);
dot(inv_image_pt,inverse_image_pen);
dot(inv_image_pt,inverse_image_pen+gray(1-colorfactor));
}
// the vector endpoint is 0.15 ps points from inverse image dot
domain_vec[i]=(0,0,0)--(xpart(plane_endpt[i]),ypart(plane_endpt[i]),starting_hgt[i]/6);
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=2.25cm; size(0,height);
import settings;
settings.render=-3;
settings.render=0; // for png: -3;
settings.maxtile=(20,20);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-7; // fewer jaggies but very slow
settings.render=0; // for png: -7; // fewer jaggies but very slow
settings.maxtile=(20,20);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-7; // fewer jaggies but very slow
settings.render=0; // for png: -7; // fewer jaggies but very slow
settings.maxtile=(20,20);
import three;
......
......@@ -2,7 +2,7 @@
import jh;
real height; height=3cm; size(0,height);
import settings;
settings.render=-10; // fewer jaggies but very slow
settings.render=0; // for jpg: -10; // fewer jaggies but very slow
settings.maxtile=(10,10);
import three;
......
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......@@ -185,8 +185,8 @@ with positive size.
\vec{v}_2=\colvec{-2 \\ 3 \\ 1}
%\vec{v}_3=\colvec{0 \\ -1 \\ 2}
\qquad
\vcenteredhbox{\only<1>{\includegraphics{asy/four_ii_orientation.png}}%
\only<2->{\includegraphics{asy/four_ii_orientation_pos.png}}}
\vcenteredhbox{\only<1>{\includegraphics{asy/four_ii_orientation.pdf}}%
\only<2->{\includegraphics{asy/four_ii_orientation_pos.pdf}}}
\end{equation*}
Those two vectors span a plane, which divides three-space in two.
The $\vec{v}_3$ shown is on the side of the plane containing vectors
......@@ -221,14 +221,14 @@ give a negative determinant.
1 &1 &-2
\end{vmat}=-25
\end{equation*}
\only<1>{\centergraphic{asy/four_ii_orientation_neg0.png}}%
\only<2>{\centergraphic{asy/four_ii_orientation_neg1.png}}%
\only<3>{\centergraphic{asy/four_ii_orientation_neg2.png}}%
\only<4>{\centergraphic{asy/four_ii_orientation_neg3.png}}%
\only<5>{\centergraphic{asy/four_ii_orientation_neg4.png}}%
\only<6>{\centergraphic{asy/four_ii_orientation_neg5.png}}%
\only<7>{\centergraphic{asy/four_ii_orientation_neg6.png}}%
\only<8->{\centergraphic{asy/four_ii_orientation_neg7.png}}%
\only<1>{\centergraphic{asy/four_ii_orientation_neg0.pdf}}%
\only<2>{\centergraphic{asy/four_ii_orientation_neg1.pdf}}%
\only<3>{\centergraphic{asy/four_ii_orientation_neg2.pdf}}%
\only<4>{\centergraphic{asy/four_ii_orientation_neg3.pdf}}%
\only<5>{\centergraphic{asy/four_ii_orientation_neg4.pdf}}%
\only<6>{\centergraphic{asy/four_ii_orientation_neg5.pdf}}%
\only<7>{\centergraphic{asy/four_ii_orientation_neg6.pdf}}%
\only<8->{\centergraphic{asy/four_ii_orientation_neg7.pdf}}%
\end{frame}
......
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......@@ -149,15 +149,11 @@ then
eval $result
done
# Create all the graphics that will be .png files
echo "================="
echo " creating png graphics with Asymptote"
echo " (some of these take minutes for a single file to compile)"
echo "================="
declare -a png_out_files=("three_ii_3dproj1" "three_ii_3dproj2" "three_ii_3dproj3" "three_ii_dims" "three_vi_3dprojtoline" "three_vi_3dgramschmidt0" "three_vi_3dgramschmidt1" "four_ii_orientation" "four_ii_orientation_pos" "four_ii_orientation_neg")
cmd="asy -f png filename" # will substitute for "filename"
# Converting over from png to pdf without 3d stuff
declare -a pdf_out_files=("three_ii_3dproj1" "three_ii_3dproj2" "three_ii_3dproj3" "three_ii_dims" "three_vi_3dprojtoline" "three_vi_3dgramschmidt0" "three_vi_3dgramschmidt1" "four_ii_orientation" "four_ii_orientation_pos" "four_ii_orientation_neg")
cmd="asy -f pdf filename" # will substitute for "filename"
for i in ${png_out_files[@]}
for i in ${pdf_out_files[@]}
do
result=${cmd//filename/$i}
# echo $i
......@@ -165,6 +161,23 @@ then
eval $result
done
# # Create all the graphics that will be .png files
# echo "================="
# echo " creating png graphics with Asymptote"
# echo " (some of these take minutes for a single file to compile)"
# echo "================="
# declare -a png_out_files=("three_ii_3dproj1" "three_ii_3dproj2" "three_ii_3dproj3" "three_ii_dims" "three_vi_3dprojtoline" "three_vi_3dgramschmidt0" "three_vi_3dgramschmidt1" "four_ii_orientation" "four_ii_orientation_pos" "four_ii_orientation_neg")
# cmd="asy -f png filename" # will substitute for "filename"
# for i in ${png_out_files[@]}
# do
# result=${cmd//filename/$i}
# # echo $i
# echo $result
# eval $result
# done
# Done making the graphics
cd ..
fi
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......@@ -120,7 +120,7 @@
\newcommand{\bookref}[1]{\reff*{#1}}
\newcommand{\prewordspace}{.15em}
\newcommand{\postwordspace}{0em plus.1em minus.0em}
\newcommand{\postwordspace}{0.15em plus.1em minus.1em}
% Text headers: Example, Definition, etc.
% Each comes with an optional argument into which you can put a
......
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......@@ -562,7 +562,7 @@ every vector $\vec{v}$ has a unique representation with respect to that basis.
\ex
The plane $2x-y+z=0$ through the origin in $\Re^3$ is a vector space.
Considering that a one-equation linear system
and paramatrizing with the free variables
and parametrizing with the free variables
\begin{equation*}
P=\set{\colvec{x \\ y \\ z}
=\colvec{1/2 \\ 1 \\ 0}y
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......@@ -115,7 +115,7 @@ is this vector.
\end{frame}
\begin{frame}
\noindent This picture of that projection
\centergraphic{asy/three_vi_3dprojtoline.png}
\centergraphic{asy/three_vi_3dprojtoline.pdf}
brings out that the projection vector is quite short:
$\norm{\vec{v}}=\sqrt(6)\approx 2.45$ while
$\norm{\proj{\vec{v}}{\spanof{\vec{s}\,}}}=\sqrt{1/6}\approx 0.41$.
......@@ -253,8 +253,8 @@ The members of this basis are mutually orthogonal.
\colvec{4/3 \\ 4/3 \\ -4/3}}$
\hspace{.25in}
\vcenteredhbox{%
\only<2>{\includegraphics{asy/three_vi_3dgramschmidt0.png}}%
\only<3->{\includegraphics{asy/three_vi_3dgramschmidt1.png}}%
\only<2>{\includegraphics{asy/three_vi_3dgramschmidt0.pdf}}%
\only<3->{\includegraphics{asy/three_vi_3dgramschmidt1.pdf}}%
}\pause
\end{center}
\pause
......
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