Math 220 Project
Using Maple® For Linear Algebra
The Deadline: This project is due in my e-mail inbox on Monday,
December 7th before the beginning of class. Projects arriving after the
beginning of class will be late and will receive half credit. Late projects
will be accepted until 9:00am on Tuesday, December 8th. NO late
projects will be accepted after 9:00am on Tuesday, December 8th.
The Points: This project is worth 200 points.
The Partners: In order to be eligible for full credit, you must do this project
with one or two other people. You may get help from anyone. If you get
help from someone (even if it is another member of the class), you must
give that person credit at the end of your paper. You will not be penalized
for getting help, but if you do not acknowledge the help you get, it is
plagiarism.
The Grading: Your grade will be based on
 the accuracy of your answers,
 the precision of your vocabulary and notation,
 the clarity of your explanations,
 the professional appearance, and
 the correctness of your use of the English language.
The Idea: The more you work with mathematics and sciences, the more
likely you will be to need to publish your work in a professional way.
Getting used to Maple® now will help you use it with confidence in the
future.
What to Turn In: Turn in one Maple® worksheet as an attachment to your
e-mail. Make sure the e-mail contains your name and your partner(s)’s
name(s) in the body of the e-mail.
The worksheet should have your name and your partner(s)’s names at the
top.
1
The Assignment:
1. Type the title of the project in text: “Using Maple® For Linear Algebra”.
Then type the name of each person in your group on a separate line
below the title.
2. In all of the following instructions, the quotation marks indicate what
you should type. Do NOT type the quotation marks themselves. Press
Enter after each semi-colon or colon.
Type “with(linalg);” at the Maple prompt. This adds the
LinearAlgebra package so you can work with matrices and vectors.
Type “M := Matrix( [ [1,2,3], [4,5,6] ]);”.
Type “M;”.
Type “evalm(M);”
Type “N:= <<1,2,3> | <4,5,6>>;”
Now insert some text to explain the difference between the way M was
defined and the way N was defined.
Using the Maple® prompt again, type “N[3,2]:= -7;”, then insert some text
to explain what happened.
Using the Maple® prompt again, type “v:= <1,2,4,8>;”
Type “v[4]:=16;”
Type “v;”, then insert some text to explain what happened.
Type “evalm(v);”
3. Definition: the rank of a matrix A is defined to be the number of pivot
columns in A.
a) One way to find the rank is to calculate the reduced echelon form
of the matrix, then count the number of pivot columns.
1
4
A
Define the matrix
6

1
2 3 4
5 6 7 
®
9 12 15 in Maple , then use the rref

1 1 1
command in Maple® to find the reduced echelon form of A. Count the
pivot columns; then insert some text telling the rank of A.
b) A faster way to find the rank of a matrix using Maple® is to use
the rank command. Type “rank(A);” at the Maple® prompt.
2
4. Recall that D is a reserved name in Maple®. To avoid problems, the
matrix D is stored under the name DD.
Define the following matrices, and use the two methods described above
to find the rank of each. You will need to show the row echelon form and
the rank of each matrix, but you will not need any text to explain what you
are doing.
1
B  2
4
1 2  3 0
C  1 2
1 1
3 6  5 1
2 4
 1 1
1 2




E




1
1
1
1
4
1
2
0
4
7
1
1
3
4
1
2
10
20
13
23
1
5
3
35
38
1
6
4
56
59
1
4
D
7

5
2 3
5 6
8 9

7 9
1 
6 
21 

56 
126 

252
5. Read the discussion of linear independence in Section 1.7. Type the
definition of a linearly independent set of vectors here, using Maple® to
put in correct notation as needed.
6. Let A = [v1 v2 … vk] be a matrix whose columns are v1, v2, … , vk.
Explain why the following are logically equivalent. Use Maple® to put in
correct notation as needed.
1) The set of vectors { v1, v2, … , vk } is linearly independent.
2) The rank of A is k.
7. Use the idea of rank to explain why the set of columns of A could not
be linearly independent if A has more columns than rows. Use Maple® to
put in correct notation as needed.
3
8. Examine the matrices B, C, D, and E from Number 4. For which of
these matrices is the set of its columns a linearly independent set? You
should be able to explain your decisions without doing any more
calculations than you have already shown.
9. Use Maple® and the concept of rank to tell whether the following sets
are linearly independent or linearly dependent.
a)
2
3
v1   
5  ,
 
1 
1 
1 
v2   
  2 ,
 
9 
 3
 4
v3   
0 
 
0 
b)
 1 
 
  4
 ,
6 
 

 1 
2
5
 ,
9
 
11
3
6
 ,
12
 
1
 4 
 7 
 

15 
 
 1 
10. Write a brief conclusion explaining what you learned. Include any
advice you would give to someone working with the LinearAlgebra
package in Maple® for the first time.
4