Appendix

## 12   Appendix

### 12.1   Mathematics terminology

• dot product : The dot product or scalar product between two vectors is formed by matching up corresponding coordinates, multiplying then adding. For example
(1,2,3) · (4,5,6) = 1(4) + 2(5) + 3(6) = 32.
In MATLAB this could be done with the following commands:
>> a = [1,2,3]; b = [4,5,6];    % define two vectors
>> dot(a,b)                     % returns 32

Which uses the dot command. Alternatively we could have used matrix multiplication
>> a * b'                       % using matrix multiplication

• cross product : The cross product between two three-dimensional vectors u and v is a vector with direction given by the right-hand rule and magnitude given by ||u||||v||sinq. In MATLAB it is found with the cross command
>> u = [1,2,3]; v = [4,5,6];    % two vectors
>> w = cross(u,v)               % the cross product
ans = -3 6 -3
>> dot(u,w)                     % cross product is orthogonal to u and v
ans = 0

• scalar : In Linear Algebra, a scalar is a name for a real number.

• matrices and a matrix : In Linear Algebra a matrix is a n × m tabular array of numbers with n rows and m columns. Matrices are quite useful for things such as keeping track of systems of equations, keeping track of connections between nodes on a graph, or even representing operations of a robot arm such as rotations. In MATLAB we can enter a matrix quite easily. For example this enters in a 2 by 3 matrix
>> [1, 2, 3; 4, 5, 6]           % the matrix [1 2 3]
%            [4 5 6]

• matrix multiplication : In Linear Algebra the product between two matrices A and B are defined if A is n by m and B is m by p. Notationally it is given by
(AB)i,j =  n å k=1
Ai,k Bk,j.
MATLAB uses the usual multiplication symbol * for matrix multiplication when the two quantities are matrices.
>> A = [1,2;3,4]                % A is 2 by 2
>> B = [5,6,7;8,9,10]           % B is 2 by 3
>> A*B                          % answer is 2 by 3
ans=
21  24  27
47  54  61
>> B*A
Warning, Warning Will Rogers    % Who said you could do this! Wrong sizes

• transpose : in Linear Algebra the transpose operator changes a n by m matrix into an m by n matrix by switching the indices. In MATLAB it is indicated with the ' sign. A typical example is to transform a row vector into a column vector.
>> v = [1,2,3]                  % a row vector
>> v'                           % a column vector

• vector or list : In Linear Algebra a vector is a list of numbers of a certain size. It can be a row vector such as
>> [1, 2, 3]                    % the vector (1,2,3)

or a column vector, which in MATLAB is entered in with
>> [1;2;3]                      % the vector (1)
%            (2)
%            (3)

In Physics, a vector is a mathematical quantity that represents a magnitude and a direction. A vector becomes a list of numbers once we chose a coordinate system, and identify a vector with the directed line segment from the origin to a point. Often the distinction between a column vector and a row vector is not made, but it is important to MATLAB.

### 12.2   Calculus terms

Here are some familiar terms from Calculus:
• continuity . Contiuity of a function is the property that the function has a limit, and the limit has the proper value. The limit as x goes to c exists and is f(c). A discontinuous function is one where either the limit does not exist (for example sin(1/x) at x=0) or when the limit does not have the correct value.

• plane . A plane is made up of all solutions to a linear equation in 3 unknowns. The general form of a plane is
a x + b y + c z = d.
This plane is normal to the vector á a,b,c ñ.

• Vector-valued functions . A function whose range is a subset of Rn, n>1.

• Asymptotes . An asymptote is a way of describing how a graph looks when an infinity is approached. A horizontal asymptote for example tells us the behaviour of a function as x = -¥,a nd x = ¥ are approached.

• Sequences . Technically, a sequence is just a funciton whose domain is a subset of the integers, usuall the positive integers. An example is the harmonic sequence 1,1/2,1/3,....

• Domain of a function . The domain of a function consists of all values for which the function is defined.

### 12.3   Some internet miscellania

Here are some definitions of some terms found in this tutorial and on the internet:

• LaTeX : LATEXis the standard typesetting program for mathematicians throughout the world. It was used to make this manual. If you are curious, and a bit adventuresome, you can find out more at http://www.tug.org

• UNIX : UNIX is an older and much more mature operating system that Windows 95 or Windows NT. To find out more, you may be interested in a free implementation for personal computers, called linux . For more information see http://www.redhat.com.

• emacs : Emacs is an editor/kitchen sink prevalent in the UNIX world. There are versions available for Windows 95, Windows NT and Windows *. For information, you can try the XEmacs home page at http://www.xemacs.org.

• linux : a free implementation of the UNIX operating system for personal computers. (As well, powerPC MacIntosh's and some workstations.) For more information you can try http://www.redhat.com.

• postscript : Postscript if a computer language that controls many printers. Consequently it is the default way to share files among internet users. In order to view or print postscript you need an interpreter (if you are a windows user). You can find one at

• m-files : An m-file'' in MATLAB is a way to store functions for later use. MATLAB stores functions in files which have the file extension .m, for example test.m, factorial.m, etc. An m-file'' needs to have a special syntax. You can read more in the section on m-files .