% help %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

help plot;
helpwin plot;
doc plot;

% matrix operations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A  =  [3 2; 2 4]
b  =  [4; 3]
c  = A*b

A = A^2;
b = b.^2;
x  = A\b % solving linear equations 

b_test(1,1) = A(1,1)*x(1) + A(1,2)*x(2);
b_test(2,1) = A(2,1)*x(1) + A(2,2)*x(2);

b
b_test


% HW0.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Problem 1
sprintf('Problem 1')

A=[-7, 1, 3; 0, -1, 0; 3, 0, 1];    % Coefficient matrix
b=[4; -2; 6];                       % RHS
x=A\b;                              % x=inverse(A) * B
sprintf('(a) The vector x is [%d, %d, %d]', x)
sprintf('(b) The determinant of A is %d.', det(A))

pause;

% Problem 2
sprintf('Problem 2')

A=[-3, -3, -1; 0, -1, 0; 3, 1, 1];
sprintf('(a) The determinant of A is %d.', det(A))

v=null(A'); v=v/v(1); 
sprintf('The rank of A is %g.', rank(A))
sprintf('The null space of transpose of A is [%d, %g, %g]', v)
sprintf('In general, there is no solution. There exists infinite solutions if [%d, %g, %g] x b = 0', v)

A=[-4, 1, 3; 0, -1, 0; 3, 0, 1; 1, 0, 0];
v=null(A'); v=v/v(1);
sprintf('The rank of A is %g.', rank(A))
sprintf('The null space of transpose of A is [%d, %g, %g, %g]', v)
sprintf('In general, there is no solution. There exists one solution only if [%g, %g, %g, %g] x b = 0', v)


pause;

%Problem 3
sprintf('Problem 3')
sprintf('(a) Columns of A are not linearly independent')
sprintf('(b) Columns of A are linearly independent')

% Problem 4
sprintf('Problem 4')

A=[-3, -3, -1; 0, -1, 0; 3, 1, 1];
v=null(A); v=v/v(1); 
sprintf('(a) The null space of  A is [%d, %g, %g]', v)

sprintf('An orthogonal basis for the range space of  A is ') 
R=orth(A) 

A=[-4, 1, 3; 0, -1, 0; 3, 0, 1; 1, 0, 0];
 
sprintf('(b) The null space of  A is ')
v=null(A)

sprintf('An orthogonal basis for the range space of  A is ') 
R=orth(A) 

A=[-4, 1, 3; 0, -1, 0; 3, 0, 1; 1, 0, 0]';
v=null(A); v=v/v(1); 
sprintf('(c) The null space of  A is [%d, %g, %g, %g]', v)

sprintf('An orthogonal basis for the range space of  A is ') 
R=orth(A) 




% Visualization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 2D plot
x_points = [-10 : .05 : 10];
line = 5 .* x_points;
parabola = x_points .^ 2;
exponential = exp(x_points);
absolute_value = abs(x_points);


figure;
subplot(2,2,1);
plot(x_points,line);
title('Here is the line');
subplot(2,2,2);
plot(x_points,parabola);
title('Here is the parabola');
subplot(2,2,3);
plot(x_points,exponential);
title('Here is the exponential');
subplot(2,2,4);
plot(x_points,absolute_value);
title('Here is the absolute value');

figure;
hold on;
plot(x_points,line,'r');
plot(x_points,parabola,'b');
plot(x_points,absolute_value,'m');
title('All in one!');
legend('line','parabola','absolute value')
grid on;
zoom on;

% 3D
% line plot
Z = [0 : pi/50 : 10*pi];
X = exp(-.2.*Z).*cos(Z);
Y = exp(-.2.*Z).*sin(Z);
plot3(X,Y,Z); grid on;
xlabel('x-axis'); ylabel('y-axis'); zlabel('z-axis');...
title('A little more interesting!');

% Surface plots
[X,Y] = meshgrid(-8:.5:8);
R = sqrt(X.^2 + Y.^2) + eps;
Z = sin(R)./R;
mesh(Z);
figure;
surf(Z);

% contour plots 
[X,Y,Z] = peaks;
contour(X,Y,Z,20);

% Vector fields
n = -2.0:.2:2.0;
[X,Y,Z] = peaks(n);
contour(X,Y,Z,10);
[U,V] = gradient(Z,.2);
hold on
quiver(X,Y,U,V)