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
algebra

pptfile
9/4

1.1,1.2.

pdf
9/6
1.3,1.4.
9/9
1.5,1.6.9/11
2.1pdf9/13

2.2, 2.3.  2.4

9/16
2.3  (Lecturer)9/18
  Holiday
9/20
Holiday
9/23
 2.5.2.6
pdf
9/25
2.5, 2.6. 9/27
3.1.
pdf
9/30

3.2.

 pdf
10/2
3.3.
pdf
10/4
3.4.,3.5   pdf
10/7
3.6.   pdf10/9
Holiday 10/11
4.1,4.2      pdf
10/14
4.1,4.2      pdf  10/16
4.5
pdf
10/18
4.5
pdf
10/21
midterm10/23
midterm10/25
midterm
10/28
5.1
pdf
10/30
5.211/1
5.3, 5.4. pdf
11/4
6.1,6.2.pdf11/6
6.3. pdf11/8
6.4. pdf
11/11
6.4
pdf
11/13
6.6.pdf11/15
6.7
11/18
6.8. 7.1
pdf
11/21
7.1.
11/23
7.2.
pdf
11/25
7.2.
pdf
11/27
7.3
pdf
11/29
7.4.
pdf
12/2
7.4pdf
8.2pdf
12/4
8.2
pdf
12/6
8.3
pdf
12/9
8.4pdf12/11
8.4pdf12/13
8.5
12/16
final12/18
final12/20
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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