parent 027e386f
 ... ... @@ -25,7 +25,7 @@ leaves extra space at start of first sentence. ** Add Topic on coupled oscillators and normal modes, maybe with video? * Refactor some code ** Move Range space and Null space to after the definition of Computing Linear Maps ** Change vector look? http://www.reddit.com/r/LaTeX/comments/m4lxo/both_of_these_arrows_look_wrong_to_me_pic/ ... ...
 ... ... @@ -1530,8 +1530,8 @@ the set of elements of $V$ that map to $$\vec{w}\in W$$ is the \definend{inverse image\/}\index{inverse image}% \index{function! inverse image} $h^{-1}(\vec{w})=\set{\vec{v}\in V\suchthat h(\vec{v})=\vec{w}}$. Above, the left side shows three inverse image sets. % Above, the left side shows three inverse image sets. \begin{example} Consider the projection\index{projection} ... ...
 // three_ii_proj_2d.asy // Illustrate inverse image of projection from \Re^2 to \Re^1 import jh; import settings; settings.outformat="pdf"; settings.render=0; real width; width=2cm; size(width,0); import graph; string fn = "three_ii_proj_2d"; // for output file name pen vector_pen=linecap(1) +linewidth(1.5pt); pen inverse_image_pen=linecap(1) +linewidth(1.5pt); pen parallogram_pen=linecap(1) +linewidth(0.4pt); pen inverse_image_fill_pen=linecap(1) +linewidth(1pt); // real red=230/255; real green=0.0; real blue=0.0; // pen red_color=rgb(red,green,blue); // red // pen shade_color=rgb(red,204/255,204/255); // change saturation to .2 // -------------------------- // Vector of length 2 in \Re^1 picture p; size(p,width,0); int picnum = 0; path vec=(0,0)--(2,0); draw(p,vec,vector_pen+red,arrow=Arrow(DefaultHead,VECTORHEADSIZE)); real[] xticks={-1,0,1,2,3,4,5}; // real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks),Arrows(TeXHead,size=1)); // yaxis("",-0.2,4.1,AXISPEN,LeftTicks("%",Size=2,yticks)); labelx(p,"\tiny $0$",N,(0,0)); shipout(fn+format("%02d",picnum), p); // -------------------------- // Vectors in \Re^2 associated with vector of length 2 picture p; size(p,width,0); int picnum = 1; real[] xticks={-1,0,1,2,3,4,5}; real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks)); yaxis(p,"",-0.2,4.75,AXISPEN,LeftTicks("%",Size=2,yticks)); for(int i=0; i < 5; ++i) { pair domain_vec_tip=(2,1+2*i/3.0); path domain_vec=(0,0)--domain_vec_tip; draw(p,domain_vec,VECTORPEN+red,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); } shipout(fn+format("%02d",picnum), p); // -------------------------- // Vector of length 3 in \Re^1 picture p; size(p,width,0); int picnum = 2; path vec=(0,0)--(3,0); draw(p,vec,vector_pen+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE)); real[] xticks={-1,0,1,2,3,4,5}; // real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks),Arrows(TeXHead,size=1)); // yaxis("",-0.2,4.1,AXISPEN,LeftTicks("%",Size=2,yticks)); labelx(p,"\tiny $0$",N,(0,0)); shipout(fn+format("%02d",picnum), p); // -------------------------- // Vectors in \Re^2 associated with vector of length 3 picture p; size(p,width,0); int picnum = 3; real[] xticks={-1,0,1,2,3,4,5}; real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks)); yaxis(p,"",-0.2,4.75,AXISPEN,LeftTicks("%",Size=2,yticks)); for(int i=0; i < 5; ++i) { pair domain_vec_tip=(3,1+2*i/3.0); path domain_vec=(0,0)--domain_vec_tip; draw(p,domain_vec,VECTORPEN+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); } shipout(fn+format("%02d",picnum), p); // -------------------------- // Vector of length 5 in \Re^1 picture p; size(p,width,0); int picnum = 4; path vec=(0,0)--(5,0); draw(p,vec,vector_pen+red+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE)); real[] xticks={-1,0,1,2,3,4,5}; // real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks),Arrows(TeXHead,size=1)); // yaxis("",-0.2,4.1,AXISPEN,LeftTicks("%",Size=2,yticks)); labelx(p,"\tiny $0$",N,(0,0)); shipout(fn+format("%02d",picnum), p); // -------------------------- // Vectors in \Re^2 associated with vector of length 5 picture p; size(p,width,0); int picnum = 5; real[] xticks={-1,0,1,2,3,4,5}; real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks)); yaxis(p,"",-0.2,4.75,AXISPEN,LeftTicks("%",Size=2,yticks)); for(int i=0; i < 5; ++i) { pair domain_vec_tip=(5,1+2*i/3.0); path domain_vec=(0,0)--domain_vec_tip; draw(p,domain_vec,VECTORPEN+red+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); } // -------------------------- // Sum of vectors picture p; size(p,width,0); int picnum = 6; real[] xticks={-1,0,1,2,3,4,5}; real[] yticks={1,2,3,4}; xaxis(p,"",-1.75,5.75,AXISPEN,RightTicks("%",Size=2,xticks)); yaxis(p,"",-0.2,4.75,AXISPEN,LeftTicks("%",Size=2,yticks)); pair domain_vec_tip_2=(2,0.5); pair domain_vec_tip_3=(3,3); pair domain_vec_tip_5=(5,3.5); path parallelogram = (0,0)--domain_vec_tip_2--domain_vec_tip_5--domain_vec_tip_3--cycle; // path parallelogram_25 = domain_vec_tip_2 -- domain_vec_tip_5; // path parallelogram_25_short = domain_vec_tip_2 -- point(parallelogram_25,0.75); // path parallelogram_35 = domain_vec_tip_3 -- domain_vec_tip_5; // path parallelogram_35_short = domain_vec_tip_3 -- point(parallelogram_35,0.6); // draw(p,parallelogram_25_short,parallogram_pen+gray(0.75)); // draw(p,parallelogram_35_short,parallogram_pen+gray(0.75)); fill(p,parallelogram,parallogram_pen+gray(0.95)); path domain_vec_2=(0,0)--domain_vec_tip_2; draw(p,domain_vec_2,VECTORPEN+red,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); path domain_vec_3=(0,0)--domain_vec_tip_3; draw(p,domain_vec_3,VECTORPEN+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); path domain_vec_5=(0,0)--domain_vec_tip_5; draw(p,domain_vec_5,VECTORPEN+red+blue,arrow=Arrow(DefaultHead,VECTORHEADSIZE),PenMargin(-1,1.5)); shipout(fn+format("%02d",picnum), p);
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