% Solving linear systems
clc
disp('                       Solving Linear Systems')
disp(' ')
disp('Define a matrix')
disp('>> A = [1 2 5; 4 5 6; 7 8 9]')
A = [1 2 5; 4 5 6; 7 8 9]
disp('Paused...Please hit a key to continue')
pause

disp('and a right-hand side (column) vector')
disp('>> b = [2 5 7]'' ')
b = [2 5 7]'
disp('Paused...Please hit a key to continue')
pause

disp('To solve Ax=b for x we type')
disp('>> x = A\b')
x = A\b
disp('Paused...Please hit a key to continue')
pause
disp('Check the answer')
disp('>> A*x - b')
A*x-b
disp('Paused...Please hit a key to continue')
pause

disp('Define another (singular) matrix')
disp('>> A = [1 2 3; 4 5 6; 7 8 9; 10 11 12]')
A = [1 2 3; 4 5 6; 7 8 9; 10 11 12]
disp('Paused...Please hit a key to continue')
pause

disp('>> rank(A)')
rank(A)
disp('Paused...Please hit a key to continue')
pause

disp('and a right-hand side (column) vector')
disp('>> b = [2 5 7 8]'' ')
b = [2 5 7 8]'
disp('Paused...Please hit a key to continue')
pause

disp('Now the (least-squares) solution is given by')
disp('>> x = A\b')
x = A\b
disp('Paused...Please hit a key to continue')
pause

disp('Note that x does NOT satisfy the linear system')
disp('>> A*x - b')
A*x-b