%%
N = 51;
circle_points = [cos(linspace(0,2*pi,N)); sin(linspace(0,2*pi,N))];
figure;
%% A matrix that stretches along x and squashes along y
A = diag([1/2 2])
b = [0.3;0.6]
%% A little messier example
%A = [0.5645 0.1360; -1.3494 1.4464]
%% A unit circle in x
subplot(1,2,1);
plot(circle_points(1,:), circle_points(2,:));
axis equal
axis([-2.1 2.1 -2.1 2.1]);
title('x space');
hold on;
%% The transformation of the unit circle under A
subplot(1,2,2);
bs = A * circle_points;
plot(bs(1,:), bs(2,:));
axis equal
axis([-2.1 2.1 -2.1 2.1]);
title('b space');
hold on;
%% A choice of X that corresponds to large norm(b)
x = [0;1];
subplot(1,2,1);
plot(x(1), x(2), 'r+');
%% A choice of X that corresponds to small norm(b)
x = [1;0];
subplot(1,2,1);
plot(x(1), x(2), 'r+');
%% The image b
b = A*x;
subplot(1,2,2);
plot(b(1), b(2), 'r+');
%% A set of small perturbations around x
xs = x * ones(1,N) + 0.1*circle_points;
subplot(1,2,1);
plot(xs(1,:), xs(2,:), 'r');
%% The effects of those perturbations on b
bs = A*xs;
subplot(1,2,2);
plot(bs(1,:), bs(2,:), 'r');
%% A choice of b that corresponds to large norm(x)
b = [1;0];
subplot(1,2,2);
plot(b(1), b(2), 'r+');
%% A choice of b that corresponds to small norm(x)
b = [0;1];
subplot(1,2,2);
plot(b(1), b(2), 'r+');
%% The solution x
x = A\b;
subplot(1,2,1);
plot(x(1), x(2), 'r+');
%% A set of small perturbations around b
bs = b * ones(1,N) + 0.1*circle_points;
subplot(1,2,2);
plot(b(1), b(2), 'r+');
plot(bs(1,:), bs(2,:), 'r');
%% The effects of those perturbations on x
xs = A\bs;
subplot(1,2,1);
plot(xs(1,:), xs(2,:), 'r');