Course 18.06: Linear Algebra
Course 18.06: Linear Algebra (Spring 2005)
Department of Mathematics
Massachusetts Institute of Technology
General Information
Lecturer: Gilbert Strang, room 2-240, e-mail gs@math.mit.edu
Lectures: MWF 11-12 room 54-100
Course Administrator: Damiano Testa, room 2-586, phone 3-4102, e-mail damiano@math.mit.edu.
Office hours: Monday 4-6pm
Final exam: To be scheduled within May 16-20
Textbook: Introduction to Linear Algebra, 3rd Edition by Gilbert Strang published by WellesleyCambridge Press.
Recitations:
# Time Room Instructor Office Phone Office Hours E-mail@math.mit.edu
1 M 2 2-131 A. Chan
2-588 3-4110
alicec
2 M 3 2-131 A. Chan
2-588 3-4110
alicec
3 M 3 2-132 D. Testa
2-586 3-4102
damiano
4 T 10 2-132 C.I. Kim
2-273 3-4380
ikim
http://web.mit.edu/18.06/www/ (1 of 6)2005/03/08 03:14:52 Þ.Ù
Course 18.06: Linear Algebra
5 T 11 2-132 C.I. Kim
2-273 3-4380
ikim
6 T 12 2-132 W.L. Gan 2-101 3-3299
wlgan
7
T1
2-131 C.I. Kim
ikim
8
T1
2-132 W.L. Gan 2-101 3-3299
wlgan
9
T2
2-132 W.L. Gan 2-101 3-3299
wlgan
2-273 3-4380
Course information: (ps, pdf).
Syllabus: (ps, pdf).
Basic MATLAB info:
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MATLAB on Athena
Short MATLAB Tutorial. In pdf.
Goals of the Linear Algebra Course (html).
Problem Sets
Problem Set #1 (ps, pdf)
Problem Set #2 (ps, pdf) (updated at 1:55pm Thursday February 10th)
Problem Set #3 (ps, pdf)
Problem Set #4 (ps, pdf)
Demos
READ THIS ! There are new eigenvalue applets WITH SOUND (use Flashplayer)
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eigen_sound
This 3-minute demo shows eigenvectors of 2 by 2 matrices
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Course 18.06: Linear Algebra
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power_method
Powers AnV lead toward the top eigenvalue/eigenvector
eigen_sound is also broken into 7 independent pieces
Applet 1
Applet 2
Applet 3
Applet 4
Applet 5
Applet 6
MINI-LECTURES ON EIGENVALUES (with voice explanation)
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Full Lecture (all eight together)
Or to view individually (about 2 minutes each)
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det(A-\lambda I)=0
Eigenvectors and Trace
Powers
Diagonalization
Differential Equations
Symmetry
Positive Definite
SVD
JAVA DEMOS (these are interactive, without voice explanation)
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Eigenvalues
Power method
SVD (Singular Value Decomposition)
Gaussian Elimination
Determinants
Gram-Schmidt = Orthogonalization
Inner Product of Functions
Sum of Fourier Series
Gibbs Phenomenon
Aliasing
Column Spaces
Least Squares
http://web.mit.edu/18.06/www/ (3 of 6)2005/03/08 03:14:52 Þ.Ù
Applet 7
Course 18.06: Linear Algebra
Other Information
Videos of Professor Strang's Fall 1999 Lectures
Additional MATLAB info:
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Added on 11/19/2003: Here are the pictures resulting from the best rank 1, rank 5, rank 10, rank
20 and rank 50 approximations to a 499 by 750 black-and-white intensity matrix. The
approximations were obtained by keeping the k largest singular values in the SVD. The bottom
right picture is the original one.
Added on 11/3/2003: MATLAB diary from review lecture on 11/3/2003
MATLAB Teaching Codes
Best Guide to MATLAB
Cool MATLAB demos by The Mathworks.
MATLAB Recitation Demos from 1997
The 3rd edition (2003) of the textbook is now available!!
Instructors could write directly to gs@math.mit.edu to see the new book.
It has Worked Examples and many new features: Glossary, Conceptual
Questions and "Linear Algebra in a Nutshell" will be useful to everyone.
Glossary: (ps, pdf)
Conceptual Questions for Review: (ps, pdf)
Linear Algebra in a Nutshell: (ps, pdf)
Question from Professor Ian Christie, West Virginia University:
Find unit vectors h(t) and m(t) in the direction of the hour and minute hands of a clock, where t denotes
the elapsed time in hours. If t = 0 represents noon then m(0) = h(0) = (0,1). At what time will the hands
of the clock first be perpendicular? At what time after noon will the hands first form a straight line? In
the dot product m(t) * h(t), remember that sin x sin y + cos x cos y = cos(x - y). Solution: (ps, pdf)
Multiplication by Columns! The multiplication Ax produces a combination
of the columns of A. If the vectors a1, a2, ... , an are those columns, then
Ax = x1 a1 + ... + xn an = combination of columns (in the column space!)
http://web.mit.edu/18.06/www/ (4 of 6)2005/03/08 03:14:52 Þ.Ù
Course 18.06: Linear Algebra
A summary of how the properties of different matrices are reflected in the eigenvalues/eigenvectors:
(ps, pdf).
Pascal Matrices (new article by Alan Edelman and Gilbert Strang): (ps , pdf )
An Essay by Professor Strang: Too Much Calculus: (ps , pdf )
INTERESTING DEMOS:
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Gauss-Jordan demo (9/14/98)
LU demo (9/14/98)
The Media Lab's Eigenfaces Demo
Linear Algebra Records
Projections of famous and not so famous three and four dimensional solids
Interactive least squares fitting
Exams from Previous Years
Spring '03: Exam 1, March 3, 2003: (ps, pdf). Solutions: (ps, pdf).
Spring '03: Exam 2, April 9, 2003: (ps, pdf). Solutions: (ps, pdf).
Fall '03: Final: (ps, pdf).
Fall '03: Exam 1: (ps, pdf). Solutions: (ps, pdf).
Fall 2002 Exams
Spring 2002 Exams
Fall 2001 Exams
Spring 2001 Exams
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Course 18.06: Linear Algebra
Fall 2000 Exams
Spring 2000 Exams
Fall 1999 Exams
Fall 1998 Exams
Spring 1998 Exams
Fall 1997 Exams
Spring 1997 Exams
Fall 1996 Exams
Spring 1996 Exams
More Practice Exams...
Welcome to MIT's Linear Algebra Home Page for Course 18.06!
You are visitor number
since October 1, 1996.
Copyright © 2003 Massachusetts Institute of Technology
http://web.mit.edu/18.06/www/ (6 of 6)2005/03/08 03:14:52 Þ.Ù
Table of Contents for Introduction to Linear Algebra
INTRODUCTION TO LINEAR ALGEBRA
3rd Edition
Gilbert Strang
Wellesley-Cambridge Press (June 1998)
TABLE OF CONTENTS
1
1.1
1.2
Introduction
Vectors and Matrices
Lengths and Dot Products
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Solving Linear Equations
Linear Equations
The Idea of Elimination
Elimination Using Matrices
Rules for Matrix Operations
Inverse Matrices
Elimination = Factorization: A = LU
Transposes and Permutations
3
3.1
3.2
3.3
3.4
3.5
Vector Spaces and Subspaces
Spaces of Vectors
The Nullspace of A: Solving Ax = 0
The Rank and the Row Reduced Form
The Complete Solution to Ax=b
Independence, Basis, and Dimension
http://web.mit.edu/18.06/www/ila3_toc.html (1 of 3)2005/03/08 03:15:00 Þ.Ù
Table of Contents for Introduction to Linear Algebra
3.6
Dimensions of the Four Subspaces
4
4.1
4.2
4.3
4.4
Orthogonality
Orthogonality of the Four Subspaces
Projections
Least Squares Approximations
Orthogonal Bases and Gram-Schmidt
5
5.1
5.2
5.3
Determinants
The Properties of Determinants
Permutations and Cofactors
Cramer's Rule, Inverses, and Volumes
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Eigenvalues and Eigenvectors
Introduction to Eigenvalues
Diagonalizing a Matrix
Applications to Differential Equations
Symmetric Matrices
Positive Definite Matrices
Similar Matrices
The Singular Value Decomposition
7
7.1
7.2
7.3
7.4
Linear Transformations
The Idea of a Linear Transformation
The Matrix of a Linear Transformation
Change of Basis
Diagonalization and the Pseudoinverse
http://web.mit.edu/18.06/www/ila3_toc.html (2 of 3)2005/03/08 03:15:00 Þ.Ù
Table of Contents for Introduction to Linear Algebra
8
8.1
8.2
8.3
8.4
8.5
8.6
Applications
Matrices in Engineering
Graphs and Networks
Markov Matrices and Economic Models
Linear Programming
Fourier Series: Linear Algebra for Functions
Computer Graphics
9
9.1
9.2
9.3
Numerical Linear Algebra
Gaussian Elimination in Practice
Norms and Condition Numbers
Iterative Methods for Linear Algebra
10
Complex Vectors and Complex Matrices
10.1 Complex Numbers
10.2 Hermitian and Unitary Matrices
10.3 The Fast Fourier Transform
Solutions to Selected Exercises
Conceptual Questions for Review
Glossary: A Dictionary for Linear Algebra
Index
Linear Algebra in a Nutshell
http://web.mit.edu/18.06/www/ila3_toc.html (3 of 3)2005/03/08 03:15:00 Þ.Ù
Gilbert Strang's Home Page
Gilbert Strang
Professor of Mathematics
Department of Mathematics
Massachusetts Institute of Technology
Room 2-240
Cambridge MA 02139
Phone: (617) 253-4383
Fax: (617) 253-4358
Send e-mail to: gs@math.mit.edu
Short Biography: Short Biography
Curriculum Vitae: Curriculum Vitae
Books and Publications
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Complete List of Publications
Recent Wavelet Papers
Textbooks
Visit the new Wellesley-Cambridge Press website.
2000-2001 Classes
18.06 Linear Algebra (The 18.06 lectures for Fall 1999 are now on the web.)
18.085 Applied Mathematics and Engineering Mathematics
18.327 Wavelets and Filter Banks
http://www-math.mit.edu/~gs/2005/03/08 03:15:37 Þ.Ù
18.06 Linear Algebra, Spring 2005
Lecturer:
Lecture hour:
Office:
Email:
Phone:
Gilbert Strang
MWF at 11 in 54–100
2–240
gs@math.mit.edu
3–4383
Course coordinator:
Office hours:
Office:
Email:
Phone:
Damiano Testa
2–586
damiano@math.mit.edu
3–4102
* * Course Web Page: http://web.mit.edu/18.06/www/ Watch there for all messages.
Text: Introduction to Linear Algebra (Third Edition) by Gilbert Strang.
The official bookstore for 18.06 is Quantum Books (corner of Broadway and Ames), and the
Coop.
Recitations: You must enroll in a specific section (they are listed on web.mit.edu/18.06/).
Your homework and exams will go to that section. Changes are made in the Undergraduate
Math Office (2–108). Your recitation instructor (not your lecturer!!!) is the person to ask all
* * questions about homework and grades.
Homework: Assignments will be due on Wednesdays, BEFORE 4PM. Please put them in
the box for your section in 2–106, next to the Undergraduate Mathematics Office. Please
staple them (you may use the UMO stapler). They are due every week except exam weeks
and are returned in recitation. Late homework is not accepted.
The homeworks are essential in learning linear algebra. They are not a test and you are
encouraged to talk to other students about difficult problems—after you have found them
difficult. Talking about linear algebra is healthy. But you must write your own solutions.
Exams: There will be three one-hour exams (in Walker) at class times on Monday February
28, Friday April 1, and Wednesday May 4. There is a final exam which the registrar will
schedule within May 16–20. The use of calculators or notes is not permitted during the
exams.
Your grade: Problem sets 24%, three one-hour exams 42%, final exam 34%.
MATLAB: Some homework problems will require you to use MATLAB, which is available
on Athena and other systems. The course web page has more information on MATLAB,
including a tutorial. MATLAB is the outstanding software for linear algebra. 18.06 will use
it for the best homework problems. The student version of MATLAB (Prentice-Hall) is now
upgraded with great graphics. The full newest version of MATLAB is on Athena.
Videos: Videos of Professor Strang’s lectures in an earlier year are available on the course
web page and also ocw.mit.edu. The VCR format is in the Barker Engineering library.
Syllabus for 18.06 Linear Algebra, Spring 2005
MWF 11–12
Room 54-100
The three midterm exams will be held in Walker during lecture hours:
closed book. All grading is by your recitation instructor!
W
F
M
W
F
M
W
F
T
W
F
M
W
F
M
W
F
M
W
F
M-F
M
W
F
M
W
F
M
W
F
M
W
F
M
W
F
M
W
F
M
W
M-F
2/2
2/4
2/7
2/9
2/11
2/14
2/16
2/18
2/22
2/23
2/25
2/28
3/2
3/4
3/7
3/9
3/11
3/14
3/16
3/18
3/21-25
3/28
3/30
4/1
4/4
4/6
4/8
4/11
4/13
4/15
4/18
4/20
4/22
4/25
4/27
4/29
5/2
5/4
5/6
5/9
5/11
5/16-20
The Geometry of Linear Equations
Elimination with Matrices
Matrix Operations and Inverses
LU and LDU Factorization
Transposes and Permutations
Vector Spaces and Subspaces
The Nullspace: Solving Ax = 0
Rectangular P A = LU and Ax = b
Row Reduced Echelon Form
Basis and Dimension
The Four Fundamental Subspaces
Exam 1: Chapters 1 to 3.5
Graphs and Networks
Orthogonality
Projections and Subspaces
Least Squares Approximations
Gram-Schmidt and A = QR
Properties of Determinants
Formulas for Determinants
Applications of Determinants
SPRING BREAK
Eigenvalues and Eigenvectors
Exam review
Exam 2: Chapters 1–5
Diagonalization
Markov Matrices
Fourier Series and Complex Matrices
Differential Equations
Symmetric Matrices
Positive Definite Matrices
PATRIOTS DAY
Matrices in Engineering
Singular Value Decomposition
Similar Matrices
Linear Transformations
Choice of Basis
Exam Review
Exam 3: Chapters 1–8 (8.1, 2, 3, 5)
Fast Fourier Transform
Linear Programming
Numerical Linear Algebra
Final Exams
1.1–2.1
2.2–2.3
2.4–2.5
2.6
2.7
3.1
3.2
3.3–3.4
3.3–3.4
3.5
3.6
8.2
4.1
4.2
4.3
4.4
5.1
5.2
5.3
6.1
6.2
8.3
8.5, 10.2
6.3
6.4
6.5
8.1
6.7
6.6
7.1–7.2
7.3–7.4
10.3
8.4
9.1–9.3
MATLAB on Athena (AC-71) Table of Contents
MATLAB on Athena (AC-71)
Table of Contents
(Answers to Common Questions About Matlab)
Getting Started
... Starting a Session
... Running a MATLAB Demo
Matrix/Vector Operations
... Creating and Working with Matrices
... Basic Arithmetic
... Element-Wise Operations
... Logical Operations
... Control Structures
... Selective Indexing
... Polynomial Operations
... Signal Processing Functions
... Libraries and Search Paths
Graphics
... Plotting Individual Graphs
... Plotting Multiple Graphs
... Output of Plots and Graphs
... ... Sending Graphics to a Printer
... ... Sending Graphics to a File
Creating Your Own Functions and Scripts
... Creating Your Own Functions
... Functions Returning No Values
... Functions Returning One Value
... Functions Returning More Than One Value
... Functions Taking a Variable Number of Arguments
... Script M-files
Saving Your Work
http://web.mit.edu/olh/Matlab/TOC.html (1 of 2)2005/03/08 03:18:35 Þ.Ù
MATLAB on Athena (AC-71) Table of Contents
Interface Controls
... Controlling the MATLAB Session
... Aspect Ratio Control
... Working With the Operating System
Modeling Dynamic Systems (SIMULINK)
For More Help about MATLAB
... Online Help
... Hardcopy Documentation
http://web.mit.edu/olh/Matlab/TOC.html (2 of 2)2005/03/08 03:18:35 Þ.Ù
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Course 18.06: Linear Algebra
Goals of the Linear Algebra Course
The goals for 18.06 are using matrices and also understanding them. Here are key computations and
some of the ideas behind them:
1. Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility
of A, factorization into A = LU)
2. Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special
solutions to Ax = 0 from row reduced R)
3. Basis and dimension (bases for the four fundamental subspaces)
4. Least squares solutions (closest line by understanding projections)
5. Orthogonalization by Gram-Schmidt (factorization into A = QR)
6. Properties of determinants (leading to the cofactor formula and the sum over all n! permutations,
applications to inv(A) and volume)
7. Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k and matrix exponentials to
solve difference and differential equations)
8. Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests
for x'Ax > 0, applications)
9. Linear transformations and change of basis (connected to the Singular Value Decomposition -orthonormal bases that diagonalize A)
10. Linear algebra in engineering (graphs and networks, Markov matrices, Fourier matrix, Fast Fourier
Transform, linear programming)
http://web.mit.edu/18.06/www/Fall02/goals.html2005/03/08 03:19:17 Þ.Ù
18.06 - Spring 2005 - Problem Set 1
February 10, 2005
This problem set on lectures 1 – 3 is due Wednesday (February 9th), at 4 PM,
in 2-106. Make sure to include your name and recitation number in your
homework! The numbers of the sections and exercises refer to “Introduction to
Linear Algebra, 3rd Edition, by Gilbert Strang.”
Please staple your solution as first page of your homework. Remember to
PRINT your name, Recitation number and Instructor name.
Lecture 1:
• Read: book sections 1.1 to 2.1.
• Work: book section 1.1 (exercise 28), 1.2 (exercise 29 and 31), and
2.1 (exercises 18 and 19).
Lecture 2:
• Read: book sections 2.2 and 2.3.
• Work: book section 2.2 (exercises 5, 7, 15, 19 and 26), and 2.3
(exercises 3, 11, 19 and 27).
Lecture 3:
• Read: book sections 2.4 and 2.5.
• Work: book section 2.4 (exercises 2, 24, 33).
Challenge Problem for 3 by 3 systems Ax = b
Success would be 3 columns of A whose combinations give every vector b,
which matches with 3 planes in the row picture that intersect at one point
(the unique solution x). Give numerical examples of these two types of
failure:
• 3 columns lie on the same line.
The 3 planes in the row picture are . . . ?
3 planes are parallel
Then if b happens to lie on that line of columns, the 3 planes meet
in a . . . ?
• 3 columns in the same plane, but no two on the same line.
Then 3 planes do what? Show by a sketch. Which b’s allow Ax = b
to be solved?
1
18.06 - Spring 2005 - Problem Set 2
This problem set on lectures 4 – 6 is due Wednesday (February 16th), at 4
PM, at 2-106. Make sure to include your name and recitation number in
your homework! The numbers of the sections and exercises refer to “ Introduction to Linear Algebra, 3rd Edition, by Gilbert Strang.”.
Please staple your solution as first page of your homework. Remember to
PRINT your name, Recitation number and Instructor name.
Lecture 4:
• Read: book sections 2.5 and 2.6.
• Work: book section 2.5 (exercises 8, 23, 30, 32 and 35) and 2.6 (6,
10, 13, and 20).
Lecture 5:
• Read: book section 2.7.
• Work: book section 2.7 (exercises 4, 12, 13, 17 and 40).
Lecture 6:
• Read: book section 3.1.
• Work: book section 3.1 (exercises 5, 10, 18, 23 and 24).
Challenge Problem
−1
instead of subtract with Eij ) of an elementary
The inverse (add with Eij
elimination matrix is the identity with +`ij added in the i, j position. The
−1
magic of A = LU is that multiplying those Eij
in reverse order puts every
`ij unchanged into L.
The problem is to prove this key Lemma:
Suppose the matrix M has the `’s filled in up to but not including
(i, j), and N is the next matrix with that next `ij filled in. Both
have 1’s down the main diagonal, and columns before j are all
−1
filled in. PROVE THAT M Eij
= N.
−1
Then every Eij
fills in its `ij and the product of them all is the correct
−1
is multiplying on the right—what
lower triangular L. Notice that Eij
does that do to the columns of a matrix ?
1
18.06 - Spring 2005 - Problem Set 3
This problem set on lectures 7 – 9 is due Wednesday (February 23th), at 4
PM, at 2-106. Make sure to include your name and recitation number in
your homework! The numbers of the sections and exercises refer to “ Introduction to Linear Algebra, 3rd Edition, by Gilbert Strang.”.
Please staple your solution as first page of your homework. Remember to
PRINT your name, Recitation number and Instructor name.
Lecture 7:
• Read: book section 3.2.
• Work: book section 3.2 (exercises 9, 15, 18, 20, 23, 27, and 28).
Lecture 8:
• Read: book section 3.3.
• Work: book section 3.3 (exercises 8, 13, 17, 18, and 19).
Lecture 9:
• Read: book section 3.4.
• Work: book section 3.4 (exercises 1, 6, 10, 24 and 31).
Challenge Problem 1
Suppose R (an m × n matrix) is in row reduced echelon form
I
0
F
0
,
with r nonzero rows and first r pivot columns.
a) Describe the column space and nullspace of R.
b) Do the same for the m × 2n matrix B = ( R R ).
R
c) Do the same for the 2m × n matrix C =
.
R
d) Finally, do the same for the 2m × 2n matrix D =
R R
.
R R
Challenge Problem 2
a) Suppose that A is a 3 × 3 matrix. What relation is there between the
nullspace of A and the nullspace of A2 ? How about the nullspace of
A3 ?
b) The set of polynomials of degree at most four in the variable x is a
d2
vector space. What is the nullspace of dx
2 ? What is the nullspace of
2 2
d
?
dx2
1
18.06 - Spring 2005 - Problem Set 4
This problem set is due Wednesday (March 9th), at 4 PM, at 2-106. Make
sure to PRINT your name, recitation number and instructor on your
homework!
Please staple your MATLAB solutions as first pages of your homework.
Lecture 11:
• Read: book section 3.6.
• Work: book section 3.6 (exercises 4, 25, 26 and 29)
Lecture 12:
• Read: book section 8.2.
• Work: book section 8.2 (exercises 11 and 17).
Lecture 13:
• Read: book section 4.1.
• Work: book section 4.1 (exercises 6, 7, 10, 26, 28 and 30).
Lecture 14:
• Read: book section 4.2.
• Work: book section 4.2 (exercises 4, 13, 17, 19, 27 and 29).
MATLAB Problems
Construct the following 6 × 6 matrices:
• K = toeplitz ([2, -1, zeros (1, 4)])
• T = K ; T(1, 1) = 1
• C = toeplitz ([2, -1, zeros(1, 3), -1])
1. C is singular: Explain why. If A is the incidence matrix (Sec. 8.2) for
a loop of 6 nodes and edges (a hexagon) verify by hand or MATLAB
that C = AT A.
2. The matrix T has a simple inverse inv(T ). Find a formula for the i, j
entry of T −1 when T is n × n.
3. The matrix K−T is certainly a rank one matrix. Compute T −1 −K −1
(6 × 6) and express it in the rank one form uv T . This is an important
example of Problem 2.5.43.
1
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Eigenvalue Applet with Sound -- All
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Eigenvalue Applet with Sound -- 1
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Eigenvalue Applet with Sound -- 2
Eigenvalue Applet with Sound -- 2
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Eigenvalue Applet with Sound -- 3
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Eigenvalue Applet with Sound -- 4
Eigenvalue Applet with Sound -- 4
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Eigenvalue Applet with Sound -- 5
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Eigenvalue Applet with Sound -- 6
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