Paul Halmos Finite Dimensional Vector Spaces Solution Homework
Math 2370 *** MATRICES & LINEAR OPERATORS *** Fall 2008
Instructor:David Swigon
Office: Thackeray 511, 412-624-4689, swigon@pitt.edu
Lectures: MWF 10:00-10:50am, Thackeray 704
Recitations:Th 10-10:50am, Thackeray 704, InstructorJonathan Holland jonathan.e.holland@gmail.com
Office Hours: MW 2:00am-3:30pm, Thackeray 511, or by appointment.
Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math2370.html
Course Description
Undergraduate linear algebra courses deal primarily with matrices and their properties, solution of systems of linear equations, eigenvalue analysis. This course covers the basic theory of linear vector spaces and linear transformations using axiomatic approach that builds upon primitive concepts toward more complex ones.
Prerequisites
Undergraduate linear algebra or matrix theory.
Textbook
- Lax, Linear Algebra, Wiley-Interscience, 1996. ISBN: 0471111112
A higher level mathematical monograph chosen for its axiomatic approach to the subject and its contents, which include most of the topics to be covered in the course.
(Available at The Book Center on Fifth Ave.)
Other recommended books
- Halmos, Finite-Dimensional Vector Spaces
Beautifully written book that provides brief and to-the-point explanation of the main concepts and their motivations. Should be read alongside of Lax.
- Hoffman & Kunze, Linear Algebra
Book written from the point of view of abstract algebra. Rich in examples and exercises, covers more than necessary.
- Axler, Linear Algebra Done Right
Undergraduate level, focuses on matrices, good for first reading or as a refresher.
Grading Scheme
Quizzes: 15%
Homework: 20%
Midterm Exam: 30%
Final Exam: 35%
Schedule
Tentative syllabus is given below, with a list of assigned reading for the week.
Every Friday a list of practice problems for the next week will be given. The students are expected to work out the practice problems and consult the instructor if difficulties arise. Few times during the term, with advance notice, the problems will be collected and graded as a homework. The approximate schedule and a list of practice problems will be posted below.
Every Friday the lecture will start with a 20 min quiz based on the material covered since the last quiz. Quizzes will be graded and will count towards the final grade.
The recitations on Thursdays will be spend on the discussion of practice problems assigned for the week, so please come prepared.
Syllabus
Week | Reading | Topics | Practice problems | Notes |
Aug 25 - Aug 29 | [L] 1 | Linear space, linear dependence, basis, dimension, subspace, quotient space | Set I | Quiz1_solution |
Sep 1 Labor Day Sep 2 - Sep 5 | [L] 2 | Linear functionals, annihilator codimension | Set II | |
Sep 8 - Sep 12 | [L] 3 | Linear mappings, domain, nullspace, range, fundamental theorem | Set III | Quiz3_solution |
Sep 15 - Sep 19 | [L] 3 | Algebra of linear mappings, transposition, similarity, projections, Matrices, rank | Set IV is a HOMEWORK due Sep 19 | |
Sep 22 - Sep 26 | [L] 4 | Simplices, trace, permutation group | Set V | Quiz4_solution |
Sep 29 - Oct 1 | [L] 5 | Determinant, multiplicative property | Set VI | Quiz5_solution |
Oct 6 - Oct 10 | [L] 5 | Laplace’s expansion, Cramer’s rule, trace | ||
Oct 14 | Review | |||
Oct 17 | Midterm Exam Covers [L] 1-5 | Solutions | ||
Oct 20 - Oct 24 | [L] 6 | Iteration of linear maps, eigenvalues, eigenvectors, characteristic polynomial, Spectral mapping theorem | Set VII is a HOMEWORK due Oct 24 | Quiz6_solution |
Oct 27 - Oct 29 | [L] 6 | Cayley-Hamilton theorem, Generalized eigenvectors, minimal polynomial, similarity of matrices | Set VIII | Quiz7_solution |
Nov 3 - Nov 7 | [L] 7 | Scalar product, distance, orthonormal basis, completeness, local compactness | Set IXis a HOMEWORK due Nov 12 | Quiz8_solution |
Nov 10 - Nov 14 | [L] 7 | Orthogonal complement, orthogonal projection, adjoint, norm, Isometry, orthogonal groups | Set X | |
Nov 24 Thanksgiving | [L] 8 | Quadratic forms, spectral resolution, | ||
Dec 1 - Dec 5 | [L] 8 | commuting maps, Normal maps, Rayleigh quotient, | Set XI | |
Dec 8 | [L] 8 | minmax principle | ||
Dec 10 | Review | |||
Dec 12 | Final Exam 10:00 - 12:00am Thackeray Rm 704 |
Linear algebra 212B (Fall 2013)
Time: M,W,F 10:00-11:00
Room: E6-1 3435
Lecturer: Suhyoung Choi schoi at math.kaist.ac.kr
Lecture assistant: To be announced
Grade distributions: A:30%, B:40%, C or below 30%. (I will include the people who drop the course.)
Lecturer: Suhyoung Choi at Room E6-4403
schoi at math dot kaist dot ac dot kr
This course begins abstract mathematics and is a good introduction to all the methods of
modern abstract mathematics. This course is your finest initiation into abstract mathematics which you won’t find
in any other course inthe universities today. So take this opportunity to develop this mode of
thinking. I wrote and linked some help at http://math.kaist.ac.kr/~schoi/teaching.html
This course concentrates on justifying the linear algebra theorems and procedures with
proofs, definitions and so on. You will learn to prove some theorems here.
(A part of the purpose of this course is to introduce you to proving theorems, lemmas,
and corollaries.)
NOTE TO STUDENTS: There is also Section 212A where a different philosophy of lecture is followed where more material will be covered.
You are expected to have prepared for the lecture by reading ahead and solving
some of the problems.
Course Book: Linear algebra 2nd Edition by Hoffman and Kunz Prentice Hall
Helpful references:
Paul R. Halmos, Finite dimensional vector spaces, UTM, Springer (mostly theoretical but no contain any rational canonical forms.)
B. Hartley, Rings, Modules, and Linear Algebra, Chapman and Hall
Larry Smith, Linear Algebra, 2nd Edition, Springer (Similar to our book, But fields are either the real field or
the complex field.)
Seldon, Axler, Linear algebra done right, Springer (Similar, The same field restriction as above)
S. Friedberg et al., Linear algebra 4th Edition, Prentice Hall (Most similar to our book. More
concrete. weak in theoretical side.)
Gilbert Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, MA, USA
*Korean books of almost the same content as Hoffman-Kunz:
선형대수학, 3판김응태,박승안공저, 청문각 2000
선형대수학개론, 정필용, 한성호, 청문각 2004
선형대수학,개정3판,임근빈,임동만공저, 형실출판사2006
Grades: Midterm(150pts) Final(150pts) Quiz (100pts) Class participation (50pts)
Total 450pts
Exams will be given according to the KAIST schedule. There are old exams at math.kaist.ac.kr/~schoi/teaching.html.
Quizzes: There will be a quiz almost every week. There will be one or two problems to solve
given 20 minutes. The quiz problems are very similar or identical with the homework problems.
One should prepare for them by groups of students working on homework problems together.
The homework problems are not to be turned in.
Introduction to formal mathematical proofs
Chapter 1: Linear equations
Chapter 2: Vector spaces
Chapter 3: Linear transformations
Chapter 4: Polynomials
Midterm
Chapter 5: Determinants
Chapter 6: Elementary canonical forms
Chapter 7: The rational and Jordan forms
Chapter 8: Inner product spaces.
Final
The teaching homepage:
http://math.kaist.ac.kr/~schoi/teaching.html
Course homepage: mathsci.kaist.ac.kr, math.kaist.ac.kr/~schoi/linearalg2013II.htm
Monday | Wednesday | Friday | |||
9/2 | Introduction to linear | 9/4 | 1.1,1.2. pdf | 9/6 | 1.3,1.4. |
9/9 | 1.5,1.6. | 9/11 | 2.1pdf | 9/13 | 2.2, 2.3. 2.4 |
9/16 | 2.3 (Lecturer) | 9/18 | Holiday | 9/20 | Holiday |
9/23 | 2.5.2.6 | 9/25 | 2.5, 2.6. | 9/27 | 3.1. |
9/30 | 3.2. | 10/2 | 3.3. | 10/4 | 3.4.,3.5 pdf |
10/7 | 3.6. pdf | 10/9 | Holiday | 10/11 | 4.1,4.2 pdf |
10/14 | 4.1,4.2 pdf | 10/16 | 4.5 | 10/18 | 4.5 |
10/21 | midterm | 10/23 | midterm | 10/25 | midterm |
10/28 | 5.1 | 10/30 | 5.2 | 11/1 | 5.3, 5.4. pdf |
11/4 | 6.1,6.2.pdf | 11/6 | 6.3. pdf | 11/8 | 6.4. pdf |
11/11 | 6.4 | 11/13 | 6.6.pdf | 11/15 | 6.7 |
11/18 | 6.8. 7.1 | 11/21 | 7.1. | 11/23 | 7.2. |
11/25 | 7.2. | 11/27 | 7.3 | 11/29 | 7.4. |
12/2 | 7.4pdf 8.2pdf | 12/4 | 8.2 | 12/6 | 8.3 |
12/9 | 8.4pdf | 12/11 | 8.4pdf | 12/13 | 8.5 |
12/16 | final | 12/18 | final | 12/20 | final |
I will be attempting to write revised lectures notes in ppt files and post them here.
MIT Linear algebra class (with recorded lectures)
There are many Java applets to play around here. See demos. Lectures are given by Gilbert Strang, the author
of one of the textbooks above. This is more for engineers but has many worthy advanced applied
mathematics in it. This course corresponds to the introduction to linear algebra course, one level below this one
MIT Linear algebra class (This correspond to our course but without rational forms.)
성균관대학교 선형대수학(more computer algebra oriented.)
Homework sets: Do not turn in your works but you should know how to solve these problems.
For quizzes, the teaching assistants will make problems similar to these. The best way is
to study the problems that were taught on that day.
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