<code>

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A simpler way to set a matrix with only a few nonzero elements

A = zeros(5,5);
A =

   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0

A(1,1) = 3
A =

   3   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0

A(2,3) = 5
A =

   3   0   0   0   0
   0   0   5   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0

% In a script file you would put these all together like so
A = zeros(5,5);
A(1,1) = 3;
A(2,3) = 5; 
% etc.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The colon operator: a fundamental trick in matlab

% 1:5 means make a vector of the numbers from 1 to 5
1:5 
ans =
     1     2     3     4     5

% exactly the same result as 
x = [1 2 3 4 5]
x =
     1     2     3     4     5

% vector of numbers from 21 to 31
x = 21:31
x =
    21    22    23    24    25    26    27    28    29    30    31

% vector of numbers from 21 to 31, in steps of 2
x = 21:2:31 
x =
    21    23    25    27    29    31

% vector of numbers from 0 to 1, in steps of 0.1
x = 0:0.1:1 % 0 to 1 in steps of 0.1
x =
  Columns 1 through 8
         0    0.1000    0.2000    0.3000    0.4000    0.5000    0.6000    0.7000
  Columns 9 through 11
    0.8000    0.9000    1.0000

load A.asc
A
A =
     0     1     2     3
     7     8     0     0
     7     8     1     9
     7     3     4     0

% Access an individual element of a matrix
A(2,2)
ans =
     8

% Accessing a range of elements of matrix, here the 2 & 3 rows in the 2nd column
A(2:3,2)
ans =
     8
     8

% Accessing a range of elements of matrix, here the 2nd-4th rows in the 2nd column
A(2:4,2)
ans =
     8
     8
     3

% Accessing a range of elements of matrix, here the 2nd-4th rows in the 1st-2nd columns
A(2:4,1:2)
ans =
     7     8
     7     8
     7     3

% Next: colon by itself means ALL rows or ALL columns

% all rows of A in 1st columns
A(:,1)
ans =
     0
     7
     7
     7

% all rows of A in 2nd columns
A(:,2)
ans =
     1
     8
     8
     3

A
A =
     0     1     2     3
     7     8     0     0
     7     8     1     9
     7     3     4     0

% 1st row, all columns
A(1,:) 
ans =
     0     1     2     3

% 4th row, all columns
A(4,:) 
ans =
     7     3     4     0

% Transpose

% set x to a row vector
x = A(4,:)
x =
     7     3     4     0
% transpose operator turns rows into columns, columns into rows

% transpose of row vector x is a column vector
x' 
ans =
     7
     3
     4
     0

% One use: rathe than entering right-hand-side vetor b like this
b = [4 ; 3 ; 7.2 ; 9]
b =
    4.0000
    3.0000
    7.2000
    9.0000

% enter it as a row vector then take transpose
b = [4  3  7.2  9]'
b =
    4.0000
    3.0000
    7.2000
    9.0000

% observe what transpose does to a matrix
A
A =
     0     1     2     3
     7     8     0     0
     7     8     1     9
     7     3     4     0
A'
ans =
     0     7     7     7
     1     8     8     3
     2     0     1     4
     3     0     9     0

% A couple a utility functions

A = zeros(4,4)
A =
     0     0     0     0
     0     0     0     0
     0     0     0     0
     0     0     0     0

A = ones(4,4)
A =
     1     1     1     1
     1     1     1     1
     1     1     1     1
     1     1     1     1
A = eye(4,4)
A =
     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

% "eye" is another matlab pun: "eye" mean I, the identity matrix
I = eye(4,4)
I =
     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

% Observe what multiplication by identity matrix does to a vector
x = rand(4,1)
x =
    0.6324
    0.0975
    0.2785
    0.5469

I*x
ans =
    0.6324
    0.0975
    0.2785
    0.5469

% Note I*x = x

% Random matrix 
A = rand(4,4) 
A =
    0.9575    0.9572    0.4218    0.6557
    0.9649    0.4854    0.9157    0.0357
    0.1576    0.8003    0.7922    0.8491
    0.9706    0.1419    0.9595    0.9340

% Null matrix i.e. a 0 x 0 matrix
A = [] 
A =

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plotting
x = 0:0.01:2*pi;

plot(x,sin(x))
xlabel('x')
ylabel('sin(x)')
clf()

plot(x,sin(x),'b', x,cos(x),'r')
legend('sin(x)','cos(x)')
xlabel('x')
ylabel('f(x)')
clf()

plot(x,sin(x),'b-.', x,cos(x),'r--')
clf()

x = 0:0.1:10
clf(); semilogy(x,exp(-x),'b-.')
clf(); semilogy(x,x.^3),'b-.')

                            |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Elementwise arithmetic

x = 1:5
x =
     1     2     3     4     5

x = (1:5)*2
x =
     2     4     6     8    10

x = (1:5) + 2
x =
     3     4     5     6     7

y = 10:14
y =
    10    11    12    13    14

x + y
ans =
    13    15    17    19    21


% however, some operations must be specified as elementwise using "dot" syntax

x
x =
     3     4     5     6     7

x.^2 % .^ means apply the power to each element
ans =
     9    16    25    36    49

% Note this doesn't work
x^2
Error using 'mpower'
Inputs must be a scalar and a square matrix.
To compute elementwise POWER, use POWER (.^) instead.} 

plot(x,x.^2)
</code>