>> %  Matlab can be used as a calculator.
>> 1+2*3

ans =

     7

>> ans/4

ans =

    1.7500

>> %  Matlab can store vectors (column vectors or row vectors).
>> v = [1; 2; 3; 4]

v =

     1
     2
     3
     4

>> w = [5, 6, 7, 8]

w =

     5     6     7    8 

>> %  You can refer to an entry of a vector by giving its index.
>> v(2)

ans =

     2

>> w(3)

ans =

     7

>> %  You can add two vectors of the same dimensions, but you cannot
>> %  add v and w because v is 4X1 and w is 1X4.  If you try, Matlab
>> %  will give you an error message.
>> v+w
??? Error using ==> +
Matrix dimensions must agree.

>> %  w' is the transpose of w.
>> w'

ans =

     5
     6
     7
     8

>> %  We can add v and w' using ordinary vector addition.
>> v + w'

ans =

     6
     8
    10
    12 

>> %  The following computes the sum of the entries of v.
>> v(1) + v(2) + v(3) + v(4)

ans =

    10

>> %  So does the following "for" loop.
>> sumv = 0;
>> for i=1:4, sumv = sumv + v(i); end;
>> sumv

sumv =

    10

>> %  Here we set a random vector of length n=100.
>> n = 100;
>> v_rand = randn(n,1);
>> %  Now we really need a "for" loop to sum the entries of v_rand.
>> sumv_rand = 0;
>> for i=1:n, sumv_rand = sumv_rand + v_rand(i); end;
>> sumv_rand

sumv_rand =

    4.7929

>> %  Actually, Matlab has a built-in function called "sum" to add 
>> %  the entries of a vector.  To find out about it, type "help sum".
>> help sum

 SUM Sum of elements.
    For vectors, SUM(X) is the sum of the elements of X. For
    matrices, SUM(X) is a row vector with the sum over each
    column. For N-D arrays, SUM(X) operates along the first
    non-singleton dimension.
 
    SUM(X,DIM) sums along the dimension DIM. 
 
    Example: If X = [0 1 2
                     3 4 5]
 
    then sum(X,1) is [3 5 7] and sum(X,2) is [ 3
                                              12];
 
    See also PROD, CUMSUM, DIFF.

>> sum(v_rand)

ans =

    4.7929

>> %  Matlab can also store matrices.
>> A = [1, 2, 3; 4, 5, 6; 7, 8, 0]

A =

     1     2     3
     4     5     6
     7     8     0

>> b = [0; 1; 2]

b =

     0
     1
     2

>> %  Matlab has lots of built-in functions for solving matrix
>> %  problems.  For example, to solve the linear system Ax=b,
>> %  type "A\b".
>> x = A\b

x =

    0.6667
   -0.3333
    0.0000

>> %  We can check this answer by computing b - A*x.  Note that
>> %  when we type A*x, this means the usual matrix-vector multiplication,
>> %  and when we subtract the result from b, this is a vector 
>> %  subtraction.  This should give a vector of 0's, if x solves
>> %  the system exactly.  Since our machine carries only about
>> %  16 decimal places, we do not expect it to be exactly zero,
>> %  but, as we will see later, it should be of size about 1.e-16.
>> b - A*x

ans =

   1.0e-15 *

   -0.0740
   -0.2220
         0

>> %  Not bad!
>> exit