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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 = cos(-(t-pi/2));

e1y = sin(-(t-pi/2));

e2x = cos(t);

e2y = sin(t);

m = [e1x e2x; ...

     e1y e2y];

m = (1+0.05*n) * m;  

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';

e1 = [0 0;1 0];

e2 = [0 0;0 1];

e1 = e1';

e2 = e2';

x = x';

tx = m * x;

taxList = m * axList;

te1 = m * e1;

te2 = m * e2;

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

qx = e1(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','red','LineWidth',2,'MaxHeadSize',0.15);

qx = e2(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','blue','LineWidth',2,'MaxHeadSize',0.15);

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

qx = te1(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','red','LineWidth',2,'MaxHeadSize',0.15);

qx = te2(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','blue','LineWidth',2,'MaxHeadSize',0.15);

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);

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

end

qx = te1(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','red','LineWidth',2,'MaxHeadSize',0.15);

qx = te2(:,1:2);

quiver(qx(1),qx(2),qx(3),qx(4),'Color','blue','LineWidth',2,'MaxHeadSize',0.15);

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

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

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

tStr = sprintf('[C] : transformed coordinate \nwith basis on the new coordinate');

title(tStr);

daspect([1 1]);

box on;

hold off;