Math 4242/4457 : Applied Linear Algebra/Methods
of Applied Math I (Section 30)
Fall Semester 2007, 2:30  3:20 pm, MWF,
Mechanical Engineering 108, Offered on the UNITE Distributed Learning system
Prereq2243 or 2373 or 2573; fall, spring, summer, every year)
Systems of linear equations, vector spaces, subspaces, bases, linear transformations,
matrices, determinants, eigenvalues, canonical forms, quadratic forms, applications
4 credits (Credit will not be granted if credit has been received for: MATH 4457; prereq 2243 or 2373 or 2573)
Instructor: Willard
Miller
Office: Vincent Hall 513
Office Hours: 10:1011:05 M, 13:2514:15 W, 9:059:55 F, or by
appointment
Phone: 6126247379
miller@ima.umn.edu,
miller@math.umn.edu
Prerequisites: Previous exposure to linear algebra: determinants and Cramer's rule.
Textbook: Applied Linear Algebra, by Peter J. Olver and Chehrzad Shakiban, PrenticeHall, Upper Saddle River, NJ, 2006.
Class Description: A foundation course in linear algebra, with applications. Topics include: linear transformations, vector spaces, matrix calculus, solutions of systems of linear equations, determinants, orthogonality, LDU decompositions, SVD decompositions, canonical forms. Applications include: GramSchmidt process, leastsquares approximations, graph theory, linear systems, etc. This material is basic, both for the understanding of the theory of linear algebra and for numerical computation.
Software: MATLAB (by The Mathworks, Inc.) will be used in the lectures and for some of the homework, No previous exposure is expected.
Policies:
Content and Style: Will cover most of Chapters 18. Homework assignments from the textbook, which has many well designed exercise problems. The theory will predominate, but there will be considerable attention to applications in other fields.
Student Conduct: Statement on Scholastic Conduct: Each student should read the college bulletin for the definitions and possible penalties for scholastic dishonesty. Students suspected of cheating will be reported to the Scholastic Conduct Committee.
Paper Grader: Xingjie Li lixxx835@math.umn.edu
Office: 320 VinH
Office Hours: 15:3017:00 T, 10:0011:30 Th
Phone: (612) 6250072
Introduction to MATLAB (courtesy
of Professor Peter Olver) Postscript
file PDF
file
Week 

Section  HW due  
W  SEP 05  1  Matrices and Vectors  1.11.2  
F  SEP 07  1  Gaussian Elimination  1.3  
M  SEP 10  2  Pivoting and Permutations  1.4  
W  SEP 12  2  Matrix Inverses  1.5  
F  SEP 14  2  Introduction to MATLAB 

M  SEP 17  3  Practical Linear Algebra  1.6/1.7 
1 
W  SEP 19  3  General Linear Systems  1.8 

F  SEP 21  3  Determinants  1.9 

M  SEP 24  4  Real Vector Spaces  2.1 
2 
W  SEP 26  4  Subspaces 
2.2 

F  SEP 28  4  Spans and Linear Independence  2.3 

M  OCT 01  5  Bases and Dimension  2.4 
3 
W  OCT 03  5  The Fundamental Matrix Subspaces  2.5 

F  OCT 05  5  Graphs and Incidence Matrices  2.6 

M  OCT 08  6  Inner Products  3.1 
4 
W  OCT 10  6  Inequalities  3.2 

F  OCT 12  6  Positive Definite Matrices  3.4 

M  OCT 15  7  Completing the Square  3.5 

W  OCT 17  7  Minimization of Quadratic Functions  4.1/4.2 
5 
F  OCT 19  7  Least Squares and the Closest Point  4.3 

M  OCT 22  8  Data Fitting and Interpolation  4.4 

W  OCT 24  8  Review  

F  OCT 26  8  Midterm I: Chapters 13, Chapter 4 (sections 4.14.3)  
M  OCT 29  9  Orthogonal Bases  5.1 
6 
W  OCT 31  9  The GramSchmidt Process  5.2 

F  NOV 02  9  Orthogonal Matrices  5.3 

M  NOV 05  10  Orthogonal Polynomials  5.4 

W  NOV 07  10  Orthogonal Projections and Least Squares  5.5 
7 
F  NOV 09  10  Orthogonal Subspaces  5.6 

M  NOV 12  11  Springs and Masses  6.1 

W  NOV 14  11  Springs and Masses 
6.1 

F  NOV 16  11  Linear Functions  7.1  
M  NOV 19  12  Linear Transformations  7.2  8 
W  NOV 21  12  Simple Dynamical Systems  8.1  
F  NOV 23  12  Thanksgiving Vacation  
M  NOV 26  13  Eigenvalues and Eigenvectors  8.2  
W  NOV 28  13  Review  
9 
F  NOV 30  13  Midterm II: Chapters 47  

M  DEC 03  14  Eigenvector Bases and Diagonalization  8.3  
W  DEC 05  14  Eigenvalues of Symmetric Matrices  8.4  10 
F  DEC 07  14  Singular Values  8.5  
M  DEC 10  15  Incomplete Matrices  8.6  
W  DEC 12  15  Review  
11 
M 
DEC 17 
Final Exam, 10:3012:30 in ME 108 
Homework Assignments
HW1 : Due in class: Monday, September 17
1.2.8 (a,b), 1.2.22, 1.2.23, 1.2.29, 1.3.1 (c),
1.3.1 (d), 1.3.3 (e), 1.3.17, 1.3.22 (a,h), 1.3.24, 1.3.26,
1.3.32 (c), 1.4.2 (a,b,c), 1.4.6, 1.4.9, 1.4.19 (c), 1.4.24.
HW2: Due in class: Monday, September 24
1.5.12, 1.5.18, 1.5.24 (e), 1.5.31 (e), 1.7.1 (b),
1.7.16, 1.7.21 (b), 1.7.22 (b), 1.7.24, 1.8.1 (b,c), 1.8.2 (f ), 1.8.5,
1.8.7 (c,i), 1.8.10.
HW3 Due in class: Monday, October 1
2.1.4, 2.1.6 (a), 2.1.7, 2.2.2 (a,b,f,g,h),
2.2.6 (a), 2.2.16 (a,b,d,f,h), 2.3.3 (a,b), 2.3.8 (a), 2.3.14,
2.3.21 (a,b,f,g), 2.3.22, 2.3.26, 2.3.29, 2.3.33 (a,c,f ).
HW4 Due in class: Monday, October 8
2.4.1 (b,c), 2.4.2 (a,b,d), 2.4.10, 2.4.14 (a),
2.4.19 (ac), 2.5.5 (a,d), 2.5.9, 2.5.13, 2.5.21 (b,c), 2.5.25 (d),
2.5.27 (c), 2.5.31, 2.6.1 (d), 2.6.3 (c), 2.6.4 (c), 2.6.7 (a).
HW5 Due in class: Wednesday, October 17
3.1.2 (b,e,f ), 3.1.9, 3.1.25, 3.2.3, 3.2.5, 3.2.12,
3.2.19, 3.2.21, 3.2.33, 3.4.2, 3.4.7, 3.4.15, 3.4.22 (ii,iii),
3.4.23 (iii), 3.4.25, 3.5.1 (a,d,e), 3.5.2 (c), 3.5.6 (b).
HW6 Due in class: Monday, October 29
4.2.3 (b,d,e), 4.2.5 (b,e), 4.2.7 (a), 4.2.9, 4.3.1,
4.3.2 (a), 4.3.8, 4.3.14 (b), 4.3.15 (d), 4.4.1 (b), 4.4.4, 4.4.5.
HW7 Due in class: Wednesday, November 7
4.4.15, 4.4.19, 4.4.33, 4.4.40 (a), 4.4.43, 4.4.51,
5.1.2, 5.1.5, 5.1.13 (a,b,c), 5.1.18, 5.1.22, 5.1.27, 5.2.1 (a), 5.2.6
(a,b,f ), 5.2.8 (for 5.2.1 (a)).
HW8 Due in class: Monday, November 19
5.3.1 (a,b,c), 5.3.8, 5.3.17 (a,bii,iii,c),
5.3.27 (a), 5.3.28 (ii), 5.4.1 (b), 5.4.2 (a), 5.4.11 (c), 5.5.1 (b,c),
5.5.2 (c), 5.5.4, 5.5.11 (b), 5.5.13, 5.6.1 (c), 5.6.2 (c,d),
5.6.17 (d), 5.6.20 (b).
HW9 Due in class: Wednesday, November 28
6.1.1, 6.1.3 (for 6.1.1 only), 6.1.7, 6.1.8 (a),
6.1.14, 6.1.16 (b).
HW10 Due in class: Wednesday, December 5
7.1.1 (a,b), 7.1.2 (a,c), 7.1.27 (a,e), 7.1.28, 7.1.38, 7.1.53, 7.2.1 (i,a,b), 7.2.24 (a), 7.2.25 (a).
HW11 Due in class: Wednesday, December 12
8.2.1 (d,i), 8.2.8 (b), 8.2.15 (d,i), 8.2.22, 8.2.24,
8.2.29 (a,b), 8.2.35 (a,b,c[i]), 8.3.1 (b,d,e), 8.3.16, 8.3.21 (b) ,
8.4.1 (d), 8.4.15 (d), 8.4.18, 8.5.1 (d), 8.5.2 (d), 8.5.11.
Class notes on proof of CauchySchwarz inequality
October 17 class notes on proofs of unique minima for quadratic forms with positive definite second order terms and for least squares
December 12 rough review notes for final exam
Practice midterm exam 1, with (very brief) solutions
Midterm exam 1, with solutions
Mean 69, Median 70, Standard Deviation 19
Advisory grades
85100 A
8084 A
7579 B+
7074 B
6569 B
6064 C+
5559 C
5054 C
4049 D+
3039 D
Mean 80, Median 84, Standard Deviation 16
Advisory grades
90100 A
8589 A
8084 B+
7079 B
6569 B
6064 C+
5559 C
5054 C
4049 D+
3039 D