Bayer Spence
 MATH V2010:
 Linear Algebra
 Spring 2007



This page is http://www.math.columbia.edu/~bayer/S07/LinearAlgebra


Section 003 (Bayer)
Tuesdays and Thursdays, 9:10am-10:25pm
304 Barnard Hall

Dave Bayer (x42643, 426 Mathematics, http://www.math.columbia.edu/~bayer)
Bulletin page | Directory of Classes : Mathematics
. . Student Services Online


Email

For correspondence concerning this course, please use the email address bayer@math.columbia.edu with a subject that includes the text [MATH V2010: Linear Algebra]


Office Hours


Exams

Including our final, there will be a total of four exams that count toward the course grade.

Test Day Date Time Points
Exam 1TuesdayFebruary 13in class25
Exam 2TuesdayMarch 20in class25
Exam 3TuesdayApril 17in class25
FinalThursdayMay 10
9:00am - Noon25

These dates do not coincide with any religious holiday which causes suspension of New York City's alternate side parking regulations; see NYC Parking Calendar. Please discuss other conflicts with me well in advance of the exam in question.

Master University Examination Schedule
University Academic Calendar


Texts

Elementary Linear Algebra: A Matrix Approach, by Lawrence E. Spence, Arnold J. Insel, and Stephen H. Friedberg, . Prentice Hall, 2000. ISBN-10: 0137167229, ISBN-13: 9780137167227. University bookstore ($125) or AddALL ($43 and up).
This is our primary text. All exercises and section numbers listed below are from this text.

Linear Algebra, by Dave Bayer, Projective Press, 2007.
This text is available online. It is a draft in progress, supplementing my lectures. It is aimed at math majors planning to continue with the Modern Algebra sequence, but other students may also find it useful.


Teaching Assistant


Course materials

We are using a new textbook, and I only recently adopted an exam schedule of four exams, so the exams from previous semesters will not match our schedule.

The following chart gives a rough indication of which old exam questions to study for each of our current exams: Exam Chart


A Note on Exam Grading

Our four exams will consist of a total of twenty exam questions over the course of the semester, each graded on a scale of 0 to 5. These are relative scores; I sort the entire class into six piles according to the answers for a particular problem, with the stronger answers receiving higher scores.

With rare exceptions, it is not possible to get a higher score than 3 for any wrong answer, no matter how inconsequential the arithmetic error leading to the wrong answer. It is possible to check the answer for every test question, and I am implicitly testing your ability to check the answer. With a deep understanding of linear algebra, one can look at any exam problem and see several different ways to tell that the answer is unquestionably correct. I am testing for this understanding.


Exercises

Exercises are not collected. Below, I am listing all exercises that you certainly should be able to solve. Use your judgment in deciding which exercises to actually work. It is a good idea to work as many exercises as you can find time to solve; your goal is to achieve fluency, so that you can complete exams comfortably in the time allowed.

You are also invited to look at the more advanced exercises that are not listed.

Section Exercises
1.1 Matrices and Vectors1-16
1.2 Linear Combinations, Matrix-Vector Products, and Special Matrices1-32
1.3 Systems of Linear Equations1-38
1.4 Gaussian Elimination1-32
1.6 The Span of a Set of Vectors1-36
1.7 Linear Dependence and Linear Independence1-34
2.1 Matrix Multiplication1-21
2.3 Invertibility and Elementary Matrices1-16
2.4 The Inverse of a Matrix1-22
2.6 Linear Transformations and Matrices1-34
2.7 Composition and Invertibility of Linear Transformations1-58
Exam 1
3.1 Cofactor Expansion1-40
3.2 Properties of Determinants1-36
4.1 Subspaces1-50
4.2 Basis and Dimension1-34
4.3 The Dimension of Subspaces Associated with a Matrix1-34
4.4 Coordinate Systems1-24
4.5 Matrix Representations of Linear Operations1-30
5.1 Eigenvalues and Eigenvectors1-36
5.2 The characteristic Polynomial1-54
Exam 2
5.3 Diagonalization of Matrices1-34
5.4 Diagonalization of Linear Operators1-26
6.1 The Geometry of Vectors1-29
6.2 Orthogonal Vectors1-28
6.3 Least-Squares Approximation and Orthogonal Matrices and Operators1-22
6.4 Orthogonal Matrices and Operators1-16
6.5 Symmetric Matrices1-20
Exam 3
7.1 Vector Spaces and their Subspaces1-11
7.2 Dimension and Isomorphism1-26
7.3 Linear Transformations and Matrix Representations1-10
7.4 Inner Product Spaces1-17
Final

Calendar

This calendar gives our schedule of classes and exams, and a rough indication of which sections will be covered on which days.

Monday Tuesday Wednesday Thursday Friday
15 Jan 06
16
1.1, 1.2
17
18
1.3
19
22 Jan
23
1.4
24
25
1.6, 1.7
26
29 Jan
30
2.1, 2.3
31
1 Feb
2.4
2
5 Feb
6
2.6
7
8
2.7
9
12 Feb
13
Exam 1
14
15
3.1
16
19 Feb
20
3.2
21
22
4.1
23
26 Feb
27
4.2, 4.3
28
1 Mar
4.4, 4.5
2
5 Mar
6
5.1
7
8
5.2
9
12 Mar
13
14
15
16
19 Mar
20
Exam 2
21
22
5.3
23
26 Mar
27
5.4
28
29
6.1
30
2 Apr
3
6.2
4
5
6.3
6
9 Apr
10
6.4
11
12
6.5
13
16 Apr
17
Exam 3
18
19
7.1, 7.2
20
23 Apr
24
7.3
25
26
7.4
27
30 Apr
1 May
2
3
4
7 May
8
9
10
FINAL
11