echo on


% This script file introduces MATLAB in the contest of solving a simple
% linear algebra problem.  It then explores some of MATLAB's plotting
% capabilities.  The lines that begin with % are comments to the user.
% Lines without a % are MATLAB commands that will be executed.  These
% commands could, of course, also be typed in interactively.
%
% The linear algebra problem:
%
%   What is the rank of:
%
%       |  1  7  -2  6 |
%   A = |  3  1   1  5 |
%       |  2  0   6  8 |
%       | -6  1  11  6 |
%
%

% press any key to continue
     pause


% Enter the Matrix A into MATLAB
    A = [1 7 -2 6;3 1 1 5;2 0 6 8;-6 1 11 6]

% Extract all elements of the columns of A into vectors.
% The semicolons following each line suppress the output.


    A1 = A(:,1);
    A2 = A(:,2);
    A3 = A(:,3);
    A4 = A(:,4);

% press any key to continue
    pause


% Sum the columns to show that the columns are not linearly independent.
% I could see this by inspection.

    col_sum = A1 + A2 + A3 - A4

% Since column 4 is the sum of the first three columns rank(A)
% is at most 3.  We can show that the first three columns are in fact
% linearly independent by doing row reduction on the matrix A.  This
% can be done by hand, or we can let MATLAB do the arithmetic with the
% command rref(A).

% press any key to continue
     pause

     rref(A)

% There are three linearly independent columns in A, and its rank is 3.
% of course, MATLAB could have computed this directly:


     rank_of_A = rank(A)

% press any key to continue
     pause

% MATLAB can also find the null space of A.  The command null(A)
% returns an orthonormal basis for the null space of A

     null_A = null(A)

% In this case A's null space consists of multiples of this vector

% press any key to continue
     pause

% So, by definition, multiplying this vector by A should result in zero:

   A*null_A

% Since we're doing this with floating point arithmetic, there is some
% error.  We didn't get exactly zero.  Notice that in double precision
% (MATLAB's default) the largest error is about 10^(-15).

% press any key to continue
     pause

% Plotting in MATLAB

% generate a 100 element equally spaced vector from 0 to L
    L = 1;
    x = linspace(0,L);

% We can create a matrix of eigenfunctions
% Note that MATLAB functions often accept vector arguments

    N = 3;  % number of eigenfunctions

    y = zeros(4,length(x));

% I'll have MATLAB stop echoing the commands while it executes the following
%   for loop.  Echo during loops will make you dizzy!
%
%    for count = 1:N,
%      y(count,:) = sin(count*pi*x/L);
%    end
   echo off

    for count = 1:N,
      y(count,:) = sin(count*pi*x/L);
    end

    echo on

% press any key to continue
  pause

% The n_th row of the 4 x 100 matrix y now contains sin(n*pi*x/L).  We can
% plot these against x, using different line types.

   plot( x,y(1,:)     )
   hold on   % leave on the previous plot and any other plots until "hold off"
   plot( x,y(2,:),'--')
   plot( x,y(3,:),'.-')
   hold off  % the next plot command will erase all of these plots

% add labels, titles, and grid marks
   xlabel('x   (units of L)')
   ylabel('f_n(x)')

   title('A Multiple Plot')

   legend('sin(\pix/L)','sin(2\pix/L)','sin(3\pix/L)',-1)
   grid on

   echo off