Slide4Math - Slides for Mathematics

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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

Matrix - Basis Transform

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

n = 0;

t = n * 2*pi/40 + pi/2;

e1x = 1;

e1y = 0;

e2x = cos(t);

e2y = sin(t);

m = [e1x e2x; ...

e1y e2y];

x = [];

for i = -10:10

x = [x ; [-10 i;10 i]];

end

for j = -10:10

x = [x ; [j -10;j 10]];

end

axList = [[-10 0;10 0];[0 -10;0 10]];

axList = axList';

pty1 = []; % point on +y axis

for i = 0:0.25:2;

pty1 = [pty1 ; [0 i]];

end

pty2 = []; % point on -y axis

for i = -2:0.25:0;

pty2 = [pty2 ; [0 i]];

end

ptx1 = []; % point on +x axis

for i = 0:0.25:2;

ptx1 = [ptx1 ; [i 0]];

end

ptx2 = []; % point on -x axis

for i = -2:0.25:0;

ptx2 = [ptx2 ; [i 0]];

end

ptph1 = []; % point on phase 1

for j = 0.25:0.25:2;

for i = 0.25:0.25:2;

ptph1 = [ptph1 ; [i j]];

end

ptph2 = []; % point on phase 2

for j = 0.25:0.25:2;

for i = -2:0.25:-0.25;

ptph2 = [ptph2 ; [i j]];

end

ptph3 = []; % point on phase 3

for j = -2:0.25:-0.25;

for i = -2:0.25:-0.25;

ptph3 = [ptph3 ; [i j]];

end

ptph4 = []; % point on phase 4

for j = -2:0.25:-0.25;

for i = 0.25:0.25:2;

ptph4 = [ptph4 ; [i j]];

end

x = x';

pty1 = pty1';

pty2 = pty2';

ptx1 = ptx1';

ptx2 = ptx2';

ptph1 = ptph1';

ptph2 = ptph2';

ptph3 = ptph3';

ptph4 = ptph4';

tx = m * x;

taxList = m * axList;

tpty1 = m * pty1;

tpty2 = m * pty2;

tptx1 = m * ptx1;

tptx2 = m * ptx2;

tptph1 = m * ptph1;

tptph2 = m * ptph2;

tptph3 = m * ptph3;

tptph4 = m * ptph4;

hFig = figure(1,'Position',[300 300 850 500]);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

subplot(2,3,[1 3]);

hold on;

plot([0.0],[0.0]);

x0 = 0.5-0.14;

y0 = 1.0-0.7;

text(x0+0.0,y0,"M = ",'FontSize',14,'color','black');

x0 = 0.42;

y0 = -0.7;

tStr = sprintf('%0.04f %0.04f\n%0.04f %0.04f', ...

m(1,1),m(1,2),...

m(2,1),m(2,2));

text(x0+0.005,y0+1.0,tStr,'FontSize',14,'color','black');

line([x0-0.01 x0-0.01],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

line([x0+0.20 x0+0.20],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

axis([0.0 1.01 0 1.2]);

set(gca,'Visible','off');

hold off;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

subplot(2,3,4);

hold on;

for i = 1:length(x)/2

lx = x(:,2*i-1:2*i);

line(lx(1,:),lx(2,:),'Color','black');

end

##for i = 1:length(axList)/2

## lx = axList(:,2*i-1:2*i);

## line(lx(1,:),lx(2,:),'Color','green','LineWidth',1);

##end

plot(pty1(1,:),pty1(2,:),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',3);

plot(pty2(1,:),pty2(2,:),'co','MarkerFaceColor',[0 1 1],'MarkerSize',3);

plot(ptx1(1,:),ptx1(2,:),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',3);

plot(ptx2(1,:),ptx2(2,:),'mo','MarkerFaceColor',[1 0 1],'MarkerSize',3);

plot(ptph1(1,:),ptph1(2,:),'go','MarkerFaceColor',[0 1 0],'MarkerSize',3);

plot(ptph2(1,:),ptph2(2,:),'yo','MarkerFaceColor',[1 1 0],'MarkerSize',3);

plot(ptph3(1,:),ptph3(2,:),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',3);

plot(ptph4(1,:),ptph4(2,:),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',3);

axis([-4 4 -4 4]);

set(gca,'xticklabel',[]);set(gca,'yticklabel',[]);

set(gca,'xtick',[]);set(gca,'ytick',[]);

tStr = sprintf('[A] : orignal coordinate \nwith basis vectors');

title(tStr);

daspect([1 1]);

box on;

hold off;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

subplot(2,3,5);

hold on;

for i = 1:length(x)/2

lx = x(:,2*i-1:2*i);

line(lx(1,:),lx(2,:),'Color','black');

end

##for i = 1:length(axList)/2

## lx = axList(:,2*i-1:2*i);

## line(lx(1,:),lx(2,:),'Color','green','LineWidth',2);

##end

plot(tpty1(1,:),tpty1(2,:),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',3);

plot(tpty2(1,:),tpty2(2,:),'co','MarkerFaceColor',[0 1 1],'MarkerSize',3);

plot(tptx1(1,:),tptx1(2,:),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',3);

plot(tptx2(1,:),tptx2(2,:),'mo','MarkerFaceColor',[1 0 1],'MarkerSize',3);

plot(tptph1(1,:),tptph1(2,:),'go','MarkerFaceColor',[0 1 0],'MarkerSize',3);

plot(tptph2(1,:),tptph2(2,:),'yo','MarkerFaceColor',[1 1 0],'MarkerSize',3);

plot(tptph3(1,:),tptph3(2,:),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',3);

plot(tptph4(1,:),tptph4(2,:),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',3);

tStr = sprintf('[B] : orignal coordinate \nwith colum vectors of M');

title(tStr);

axis([-4 4 -4 4]);

set(gca,'xticklabel',[]);set(gca,'yticklabel',[]);

set(gca,'xtick',[]);set(gca,'ytick',[]);

daspect([1 1]);

box on;

hold off;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

subplot(2,3,6);

hold on;

for i = 1:length(x)/2

lx = tx(:,2*i-1:2*i);

line(lx(1,:),lx(2,:),'Color','black');

end

##for i = 1:length(taxList)/2

## lx = taxList(:,2*i-1:2*i);