**
No knowledge of any specific computer program is needed
for this course, but matlab is a particularly appropriate
program for checking your work in this part of the course. **

**
I don't want to lecture on use of
matlab, but here
are some very primitive instructions on getting started
if you are in a Unix environment
and on getting the row-reduced echelon form of a matrix.
NONE OF THIS IS REQUIRED AND YOU ARE WELCOME TO USE ANY
LINEAR ALGEBRA PROGRAM YOU WANT TO CHECK YOUR WORK.
My comments are preceded by % signs. **

**
% Type matlab
at the prompt ---
% you will see something like this:
**

**
zariski:hochster% matlab
**

**
**

To get started, type one of these: helpwin, helpdesk, or demo.

For product information, type tour or visit www.mathworks.com.

>>

% The >> is a prompt. You can feed in a matrix at

% the prompt by typing

>> A = [1 2 3; 4 5 6]

% The semi-colons ; separate rows.

% Spaces or commas can be used to separate entries of

% one row. If you type in the above (and hit Return) you

% will see this:

>> A = [1 2 3; 4 5 6]

A =

1 2 3

4 5
6

>>

% Now, if you want to get the reduced row echelon form of A

% type in rref(A) at the prompt (and hit Return):

>> rref(A)

ans =

1 0
-1

0 1
2

% An alternative is to name the answer for future manipulations

% yourself, e.g. type in B = rref(A) (and hit return)

>> B = rref(A)

B =

1 0
-1

0 1
2

>>

% The product of two matrices A,B can be entered as A*B,

% and the inverse of the square matrix A as inv(A).

% det(A) produces the determinant of the

% square matrix A.

% norm(A) produces the length of the

& row or column vector A.

% A' produces the transpose of the

% matrix A, i.e. the same matrix but

% with the rows and columns interchanged.

% If A is 3 by 5 then A' is 5 by 3.

% For example if A is

% 1 2 3

% 4 5 6

% then A' is

% 1 4

% 2 5

% 3 6

% The dot product of two vectors A and B

% can be obtained by using transposes suitably

% so that the first becomes a row and the second

% a column, and then using * . E.g., for two

% rows the dot product is A*B' while for

% two columns the dot product is A'*B.

% null(A) produces a basis for the

% kernel (also called the nullspace)

% of A consisting of mutually

% perpendicular (= orthogonal) vectors

% of length 1. This means that what

% is standardly produced will only be

% correct to 4 decimal places and

% fractions are typically introduced

% even when that is not necessary.

% That's all for now.

**
**