Why use Maple for linear algebra?
� it quickly obtains the Final Form (RREF) of a matrix using
the Gauss
1) install a PC based version (7) over the Network to your PC.
This assumes you are hooked
up
to the network and therefore in the dorms. In all cases of students not being able to
install
Maple over the Network, it was found that they were
not part of the Academic LAN
nor in the SMS database, so their computer would not receive Advertised
Programs nor
have the S: drive mapped to \\argus\applications.�
You might go
to the following CCC
site� for detailed information which
will result in your
computer becoming part of the Academic LAN.
2) install from a CD if you have purchased a Student Edition of
Maple (bookstore etc)
3) start a Unix terminal session and use the text based version
of Maple. This would rule out graphics
but
in linear algebra this doesn�t amount to much
1) on your PC go to START�� then Control Panel� then Advertised
Programs Wizard
2) check off Maple 7 from the
list� (and/or anything else you want)
3) select Finish��� You should eventually end up with a
Maple7 icon on your desktop
����������� this assumes you are in a text based window and have
a Unix prompt, >, showing
����������� at the >
prompt simply enter��� >maple
����������� and you will
again� get a� >� prompt but this is from Maple, not Unix. This
is an older
����������� version of Maple but it will do all the linear algebra
problems fine, exactly the same as
����������� the newer versions.�
When finished, enter�
> quit�� to return
to Unix
�������
Part One�
Solving Systems of Equations
����������� (start Maple and try the
commands below as you read this. They are shown in blue )
������������������ >with(linalg);
��������� As in class, let�s solve
problem 21 d, page 63 of Kolman,
which is
������������������ x + 2y + 3z
� w =0
������������������ 2x + y - z
+ w = 3
������������������ x� - y� + w�����
=� -2
��������� Now create a 3x5 matrix A to
hold all 15 coefficients, which are entered equation by equation
������������������ >A:=matrix(3 ,5 ,[1, 2, 3, -1, 0 ,2 , 1, -1, 1, 3, 1, -1, 1,
0, -2]);
������������������ noting
that there are 15 entries inside brackets [����������� ]�
and also that
������������������ they
are entered row by row with a 0 if there is no term present (z in 3rd
eqn)
������������������ That�s the hard part! To
perform the Gauss Jordan algorithm, simple call the� function
������������������ gaussjord� and apply it to your matrix
������������������������������������� >gaussjord(A);
������������������ and the software will do the rest, producing the Final Form (RREF) for you:
������������������ at this point, you have
to assign arbitrary variables or decide there is no solution. Maple has only
done the computations for you. (w
would be arbitrary here and the final, scalar solution,
would be
��������������������������������������������������������� x = -w/3�
+�� 1/3
��������������������������������������������������������� y
=� w/15 +�� 4/3
���������������������������� ���������������������������� z =�� 2w/5���
� 1
��������������������������������������������������������� w� arbitrary��������� ).�
Note the signs.�
Part
Two� Matrix
Arithmetic With Maple� (see 10/29 class notes also)
��������� > with(linalg):���� * open linear algebra library
always *
��������� > A: = matrix (3,2,[1,2,3,4,5,6]);����������
*
create a matrix A *
> B: =� matrix(2 , 2, [,1,5,2]);�������������������
> C: = multiply(A,B);����������� * product of A and B *
> H:=
inverse(B);������������������ * multiplicative inverse, if
it has one *
> G:=multiply(B,B^2);������� * raise B to third power� *
Part
Three - Eigenvalues and Eigenvectors� (see notes from 11/25 also)
>with(linalg):
>A:= matrix(3,3,[2,1,0,1,2,1,0,1,2]);
> cp:=charpoly(A,x);
���������������������������������������������������������������������������� x3 � 6x2 + 10x -4
>solve(cp=0,x);
����������������������������������������������� x
= 2 , 2 +/- 21/2
or
look at the plot and see where it crosses the x axis
> plot(cp, x
= -5..5);��������������� (you need to adjust the
range of x to see the intercepts clearly)
or
use the library functions eigenvals and/or eigenvects
>� eigenvals(A);
������������������������������������� 2,
2 +/- √2
> eigenvects(A);
�������������������������������������� [2,1, {
[-1,0,1]}� ],� [2 +√2,1, { [1, √2,1] } ], [2-√2
, 1, { [1,- √2,1]) ]
����������� the last output is a little confusing to read. The 3 eigenvectors as in {brackets}�and are preceded by
the eigenvalue associated with them. The� number 1, which
appears each time, indicates that the eigenvalue was
a single root. You should get out of
this the following:
��������������������� eigenvalue 2,����� single root, eigenvector� [-1,0,1]
��������������������� eigenvalue 2 +√2,�single root, eigenvector� [1, √2,1]
��������������������� eigenvalue ��2
-√2,�single root,� eigenvector�
[1, -√2,1]
Later on we will deal with the problems of running into
double and triple roots.