# Archive of old courses

This is an archive of courses I have recently taught. Each course has a very brief summary. If an instructor would like my course materials or course webpage for use in a future semester, I am happy to provide it. But since I often re-use problems, etc., I do ask that these materials be shared with students wisely.

• MAT 535, Algebra II, Spring 2010
• MAT 200, Logic, Language and Proofs, Fall 2009
• MAT 534, Algebra I, Fall 2009
• MAT 131, Calculus I, Fall 2008
• MAT 131, Calculus I, Fall 2007
• MAT 211, Linear algebra, Fall 2007
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• Text: James Stewart, Single variable calculus, Stony Brook University edition
• Grading system: 10% problem sets and recitations, 25% Midterm I, 25% Midterm II, 40% Final Exam

Syllabus

• Week 1, September 3rd -- September 7th
Section 1.1 Four ways to represent a function, p. 11
Section 1.2 Mathematical models: A catalog of essential functions, p. 25
Appendix C Trigonometry, p. A18
LEC 1 and 2: No lecture Monday, September 3rd
• Week 2, September 10th -- September 14th
Section 1.5 Exponential functions, p. 55
Section 1.6 Inverse functions and logarithms, p. 63
Section 2.1 The tangent and velocity problems, p. 93
LEC 1: No lecture Friday, September 14th, LEC 2: No lecture Wednesday, September 12th, LEC 3: No lecture Thursday, September 13th
Problem Set 1 due in recitation.
• Week 3, September 17th -- September 21st
Section 2.2 The limit of a function, p. 98
Section 2.3 Calculating limits using the limit laws, p. 108
Section 2.4 Continuity, p. 117
Problem Set 2 due in recitation.
• Week 4, September 24th -- September 28th
Section 2.5 Limits involving infinity, p. 128
Section 2.6 Tangents, velocities, and other rates of change, p. 139
Problem Set 3 due in recitation.
• Week 5, October 1st -- October 5th
Section 2.7 Derivatives, p. 148
Section 2.8 The derivative as a function, p. 155
Section 2.9 What does f' say about f?, p. 168
Problem Set 4 due in recitation.
• Week 6, October 8th -- October 12th
Section 3.1 Derivatives of polynomials and exponential functions, p. 183
Section 3.2 The product and quotient rules, p. 193
MIDTERM 1 on Wednesday, October 10th, 8:30PM
No assigned problems to be collected this week.
• Week 7, October 15th -- October 19th
Section 3.4 Derivatives of trigonometric functions, p. 213
Section 3.5 The chain rule, p. 220
Problem Set 6 due in recitation.
• Week 8, October 22nd -- October 26th
Section 3.6 Implicit differentiation, p. 232
Section 3.7 Derivatives of logarithmic functions, p. 240
Section 3.8 Linear approximations and differentials, p. 247
Problem Set 7 due in recitation.
• Week 9, October 29th -- November 2nd
Section 4.1 Related rates, p. 263
Section 4.2 Maximum and minimum values
Problem Set 8 due in recitation.
• Week 10, November 5th -- November 9th
Section 4.3 Derivatives and the shapes of curves, p. 278
Section 4.5 Indeterminate forms and l'Hospital's rule, p. 297
MIDTERM 2 on Thursday, November 8th, 8:30PM
No assigned problems to be collected this week.
• Week 11, November 12th -- November 16th
Section 4.5 Indeterminate forms and l'Hospital's rule (continued), p. 297
Section 4.6 Optimization problems, p. 306
Problem Set 10 due in recitation.
• Week 12, November 19th -- November 23rd
Section 4.8 Newton's method, p. 322
Section 4.9 Antiderivatives, p. 327
LEC 1: No lecture Friday, November 23rd, LEC 3: No lecture Thursday, November 22nd
No assigned problems to be collected this week.
• Week 13, November 26th -- November 30th
Section 5.1 Areas and distances, p. 343
Section 5.2 The definite integral, p. 354
Problem Set 11 due in recitation.
• Week 14, December 3rd -- December 7th
Section 5.3 Evaluating definite integrals, p. 366
Section 5.4 The fundamental theorem of calculus, p. 377
Section 5.5 The substitution rule, p. 386
Problem Set 12 due in recitation.
• Week 15, December 10th -- December 14th
Section 5.5 The substitution rule (continued), p. 386
Final review
FINAL EXAM on Tuesday, December 18th 2--4:30PM

Problem sets

• Problem Set 1 is due in recitation the week of September 10th--14th.
Section 1.1: 2,5,6,7,8,19, 24,42,54
Section 1.2: 3,4,10
Appendix C: 1,4,15,30, 37
• Problem Set 2 is due in recitation the week of September 17th--21st.
Section 1.5: 1, 12, 13, 15, 18, 19, 26(a),(b),(c)
Section 1.6: 5, 9, 16, 17, 20, 21, 26, 33, 36, 45, 50
• Problem Set 3 is due in recitation the week of September 24th--28th.
Section 2.1: 5a
Section 2.2: 3, 4, 6, 8, 9, 10, 23(a).
Section 2.3: 1, 2, 4, 8, 10, 11, 12, 15, 20, 28, 32, 43.
• Problem Set 4 is due in recitation the week of October 1st--5th.
Section 2.4: 3, 4, 6, 13, 14, 16, 24, 25, 26, 30, 32, 37, 48.
Section 2.5: 2, 4, 5, 6, 7, 8, 15, 18, 20, 21, 24, 25, 27, 28, 46, 48a.
Section 2.6: 7, 8, 9, 10, 13, 20, 24.
• Problem Set 5 . Due to Midterm 1 this week, no problems are to be handed in. However, as always, you are strongly encouraged to look at all the problems, even those not to be handed in. This will give you further practice, and better prepare you for similar problems on future exams.
Section 2.7: 3,5,8,13,15,17,19,20,22,35,36.
Section 2.8: 3,4,6,21,22,24,32,34.
• Problem Set 6 is due in recitation the week of October 15th--19th.
Section 2.9: 1, 2, 4, 16, 20, 21.
Section 3.1: 1a, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 38, 44, 46, 48, 54, 56.
Section 3.2: 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 20, 29, 30, 32, 38.
• Problem Set 7 is due in recitation the week of October 22nd--26th.
Section 3.4: 2, 3, 5, 6, 7, 8, 9, 10, 12, 20(a), 27, 28, 30, 35.
Section 3.5: 2, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 29, 30.
• Problem Set 8 is due in recitation the week of October 29th--November 2nd.
Section 3.5: 69, 71.
Section 3.6: 3, 5, 6, 7, 9, 11, 12, 15, 18, 29, 32, 40, 43, 44.
Section 3.7: 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 19, 21, 27, 29, 30, 31, 33, 34, 35, 36.
Section 3.8: 6, 8, 15, 16, 28a.
• Problem Set 9 . Due to Midterm 2 this week, no problems are to be handed in. However, as always, you are strongly encouraged to look at all the problems, even those not to be handed in. This will give you further practice, and better prepare you for similar problems on exams (both this week's exam as well as the final exam).
Section 4.1: 1, 2, 3, 9, 11, 13.
Section 4.2: 1, 2, 3, 4, 5, 6, 7, 9, 25, 29, 33, 37, 41.
• Problem Set 10 is due in recitation the week of November 12th--16th.
Section 4.3: 7, 9, 11, 13, 19, 23, 27, 29, 30, 31, 32, 52.
Section 4.5: 1, 2, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 45, 47, 55.
• Problem Set 11 is due in recitation the week of November 26th--30th.
Section 4.5: 3, 4, 31, 33, 35, 37, 39
Section 4.6: 10, 13, 19, 24, 31, 43
• Problem Set 12 is due in recitation the week of December 3rd--7th.
Section 4.8: 4, 5, 9, 11, 26, 30.
Section 4.9: 1, 3, 5, 7, 9, 11, 12, 15, 17, 19, 21, 23, 25, 27, 39.
Section 5.1: 1,18,19,20.
Section 5.2: 1, 17, 18, 21, 22, 25, 32, 34, 38, 40, 41, 42, 44, 49.
• Text: Otto Bretscher, Linear algebra with applications, 3rd ed.
• Grading system: 15% problem sets and recitations, 25% Midterm I, 25% Midterm II, 35% Final Exam

Syllabus

• Week 1, September 3rd -- September 7th
Section 1.1 Introduction to linear systems, p. 1
Section 1.2 Matrices, vectors, and Gauss-Jordan elimination, p. 8
LEC 1: No lecture Monday, September 3rd
• Week 2, September 10th -- September 14th
Section 1.2 Matrices, vectors, and Gauss-Jordan elimination (continued), p. 8
Section 1.3 On the solutions of linear systems; matrix algebra, p. 26
LEC 1: No lecture Friday, September 14th, LEC 2: No lecture Thursday, September 13th
Problem Set 1 due (this is a half-problem set due at the beginning of the week because of the holidays).
• Week 3, September 17th -- September 21st
Section 2.1 Introduction to linear transformations and their inverses, p. 41
Section 2.2 Linear transformations in geometry, p. 55
Problem Set 2 due.
• Week 4, September 24th -- September 28th
Section 2.3 The inverse of a linear transformation, p. 70
Section 2.4 Matrix products, p. 79
Problem Set 3 due.
• Week 5, October 1st -- October 5th
Section 3.1 Image and kernel of a linear transformation, p. 100
Section 3.2 Subspaces of Rn; bases and linear independence, p. 112
Problem Set 4 due.
• Week 6, October 8th -- October 12th
Section 3.3 The dimension of a subspace of Rn, p. 123
Section 3.4 Coordinates, p. 136
Problem Set 5 due.
• Week 7, October 15th -- October 19th
Section 4.1 Introduction to linear spaces, p. 152
MIDTERM 1 in class on Chapters 1, 2 and 3.
Problem Set 6 due (this is a half-problem set).
• Week 8, October 22nd -- October 26th
Section 4.2 Linear transformations and isomorphisms, p. 164
Section 4.3 The matrix of a linear transformation, p. 171
Problem Set 7 due.
• Week 9, October 29th -- November 2nd
Section 5.1 Orthogonal projections and orthonormal projections, p. 186
Section 5.2 Gram-Schmidt process and QR factorization, p. 202
Problem Set 8 due.
• Week 10, November 5th -- November 9th
Section 5.3 Orthogonal transformations and orthogonal matrices, p. 209
Section 5.5 Inner product spaces, p. 232
Problem Set 9 due.
• Week 11, November 12th -- November 16th
Section 6.1 Introduction to determinants, p. 247
MIDTERM 2 in class on Chapters 4 and 5.
Problem Set 10 due (this is a half-problem set).
• Week 12, November 19th -- November 23rd
Section 6.2 Properties of the determinant, p. 261
LEC 1: No lecture Friday, November 23rd, LEC 2: No lecture Thursday, November 22nd
Problem Set 11 due (this is a half-problem set due at the beginning of the week because of the holidays).
• Week 13, November 26th -- November 30th
Section 6.3 Geometrical interpretations of the determinant; Cramer's rule, p. 275
Section 7.1 Dynamical systems and eigenvectors; an introductory example, p. 292
Problem Set 12 due.
• Week 14, December 3rd -- December 7th
Section 7.2 Finding the eigenvalues of a matrix, p. 305
Section 7.3 Finding the eigenvectors of a matrix, p. 317
Problem Set 13 due.
• Week 15, December 10th -- December 14th
Section 7.4 Diagonalization, p. 329
Final review
The FINAL EXAM is cumulative: it covers material from the entire semester.

Problem sets

• Problem Set 1 is due on Wednesday, September 12th at the beginning of lecture for LEC01 and on Tuesday, September 11th at the beginning of lecture for LEC02.

Section 1.1: 6, 12, 26, 30, 36

Typically problem sets will be due on Friday for LEC01 and on Thursday for LEC02. However this week has a holiday. Problem Set 1 is a half-problem set.

• Problem Set 2 is due on Friday, September 21st at the beginning of lecture for LEC01 and on Thursday, September 20th at the beginning of lecture for LEC02.

Section 1.2: 10, 18, 26, 36, 46

Section 1.3: 6, 14, 30, 44, 62

• Problem Set 3 is due on Friday, September 28th at the beginning of lecture for LEC01 and on Thursday, September 27th at the beginning of lecture for LEC02.

Section 2.1: 6, 10, 14, 42, 44, 48

Section 2.2: 4, 10, 26(a)--(d), 44

• Problem Set 4 is due on Friday, October 5th at the beginning of lecture for LEC01 and on Thursday, October 4th at the beginning of lecture for LEC02.

Section 2.3: 10, 12, 28, 42, 44

Section 2.4: 12, 30, 34, 50, 52

• Problem Set 5 is due on Friday, October 12th at the beginning of lecture for LEC01 and on Thursday, October 11th at the beginning of lecture for LEC02.

Section 3.1: 12, 24, 42, 50, 54

Section 3.2: 6, 16, 36, 46, 58

• Problem Set 6 is a half-problem set due on Friday, October 19th at the beginning of lecture for LEC01 and on Thursday, October 18th at the beginning of lecture for LEC02.

Section 3.3: 46, 54

Section 3.4: 56, 64, 70

• Problem Set 7 is due on Friday, October 26th at the beginning of lecture for LEC01 and on Thursday, October 25th at the beginning of lecture for LEC02.

Section 3.4: 36, 54, 62, 66

Section 4.1: 16, 20, 36, 38, 40, 42

• Problem Set 8 is due on Friday, November 2nd at the beginning of lecture for LEC01 and on Thursday, November 1st at the beginning of lecture for LEC02.

Section 4.2: 14, 28, 36 (but you do not have to decide if this is an isomorphism), 44, 70

Section 4.3: 7 (the answer is in the back of the book), 8, 42, 68 (c)--(e) only, 71

Originally Problem 71 was assigned as Problem 72 (since I thought Problem 71 was worked for you in the back of the book). Since Problem 71 is not worked for you, I decided it is more fair to ask you about Problem 71 rather than Problem 72.
Here is a hint: Compare Problem 71 in Section 4.3 with Problem 70 in Section 4.2 (where n goes to n-1 and the real numbers a1,…,an go to c0,…,cn-1). When the linear transformation T from Problem 70 is invertible, how does that help prove the "weights" \$w1,…,wn do exist? How can you find the weights, assuming you know T-1?

• Problem Set 9 is due on Friday, November 9th at the beginning of lecture for LEC01 and on Thursday, November 8th at the beginning of lecture for LEC02. This problem set has fewer theoretical problems and more computational problems. Hopefully this will be helpful in studying for next week's exam.

Section 5.1: 2, 10, 16, 26, 36

Section 5.2: 14, 28, 32, 36, 42

• Problem Set 10 is a half-problem set due on Friday, November 16th at the beginning of lecture for LEC01 and on Thursday, November 15th at the beginning of lecture for LEC02.

### ANNOUNCEMENT

Because Section 5.5 was not discussed in lecture, we will ask the grader to grade only the problems from Section 5.3. We hope that you will still attempt the problems from Section 5.5.

Section 5.3: 34, 38, 40

Section 5.5: 10, 20

• Problem Set 11 is a half-problem set due on Wednesday, November 21st at the beginning of lecture for LEC01 and on Tuesday, November 20th at the beginning of lecture for LEC02.

Section 6.1: 18, 26, 34, 38, 40

• Problem Set 12 is due on Friday, November 30th at the beginning of lecture for LEC01 and on Thursday, November 29th at the beginning of lecture for LEC02.

Section 6.1: 20, 30, 42

Section 6.2: 2, 6, 10, 26, 38, 40, 50

• Problem Set 13 is due on Friday, December 7th at the beginning of lecture for LEC01 and on Thursday, December 6th at the beginning of lecture for LEC02.

Section 6.3: 2, 4, 12, 24, 30

Section 7.1: 12, 18, 24, 36, 38

• Text: James Stewart, Single variable calculus, Stony Brook University edition
• Grading system: 10% problem sets and recitations, 25% Midterm I, 25% Midterm II, 40% Final Exam

Syllabus

• Week 1, September 2nd -- September 5th
Section 1.1 Four ways to represent a function, p. 11
Section 1.2 Mathematical models: A catalog of essential functions, p. 25
Appendix C Trigonometry, p. A18
LEC 1 and 2: No lecture Monday, September 1st.
• Week 2, September 8th -- September 12th
Section 1.5 Exponential functions, p. 55
Section 1.6 Inverse functions and logarithms, p. 63
Section 2.1 The tangent and velocity problems, p. 93
Problem Set 1 due in recitation.
• Week 3, September 15th -- September 19th
Section 2.2 The limit of a function, p. 98
Section 2.3 Calculating limits using the limit laws, p. 108
Section 2.4 Continuity, p. 117
Problem Set 2 due in recitation
• Week 4, September 22nd -- September 26th
Section 2.5 Limits involving infinity, p. 128
Section 2.6 Tangents, velocities, and other rates of change, p. 139
Problem Set 3 due in recitation.
• Week 5, September 29th -- October 3rd
Miscellaneous
LEC 1 : No lecture Wednesday, October 1st, LEC 2: No lecture this week (but attend recitation), LEC 3 : No lecture Tuesday, September 30th.
No problem set due this week.
• Week 6, October 6th -- October 10th
Section 2.7 Derivatives, p. 148
Section 2.8 The derivative as a function, p. 155
Section 2.9 What does f' say about f?, p. 168
LEC 2: No lecture Wednesday, October 8th (but attend recitation), LEC 3: No lecture Thursday, October 9th.
Problem Set 4 due in recitation.
If your recitation meets only once this week, then your problem set is due during that one meeting (even if it falls early in the week).
• Week 7, October 13th -- October 17th
Section 3.1 Derivatives of polynomials and exponential functions, p. 183
Section 3.2 The product and quotient rules, p. 193
MIDTERM 1 on Tuesday, October 14th, 8:30PM
No assigned problems to be collected this week.
• Week 8, October 20th -- October 24th
Section 3.4 Derivatives of trigonometric functions, p. 213
Section 3.5 The chain rule, p. 220
Problem Set 5 due in recitation.
• Week 9, October 27th -- October 31st
Section 3.6 Implicit differentiation, p. 232
Section 3.7 Derivatives of logarithmic functions, p. 240
Section 3.8 Linear approximations and differentials, p. 247
Problem Set 6 due in recitation.
• Week 10, November 3rd -- November 7th
Section 4.1 Related rates, p. 263
Section 4.2 Maximum and minimum values
MIDTERM 2 on Thursday, November 6th, 8:30PM
No assigned problems to be collected this week.
• Week 11, November 10th -- November 14th
Section 4.3 Derivatives and the shapes of curves, p. 278
Section 4.5 Indeterminate forms and l'Hospital's rule, p. 297
Problem Set 7 due in recitation.
• Week 12, November 17th -- November 21st
Section 4.6 Optimization problems, p. 306
Section 4.9 Antiderivatives, p. 327
Problem Set 8 due in recitation.
• Week 13, November 24th -- November 28th
Section 5.1 Areas and distances, p. 343
Section 5.2 The definite integral, p. 354
LEC 1: No lecture Friday, November 28th, LEC 3: No lecture Thursday, November 27th.
No assigned problems to be collected this week.
• Week 14, December 1st -- December 5th
Section 5.3 Evaluating definite integrals, p. 366
Section 5.4 The fundamental theorem of calculus, p. 377
Section 5.5 The substitution rule, p. 386
Problem Set 9 due in recitation.
• Week 15, December 8th -- December 12th
Section 5.5 The substitution rule (continued), p. 386
Final review
Problem Set 10 due in recitation.
• Week 16, December 15th
Final review (continued)
The final review will be held on Monday, December 15th following a Thursday lecture. This is a Lecture 3 meeting time. The instructor is Jason Starr. But all MAT 131 students are welcome to attend (up to the fire-code seating capacity).
For recitations which do not meet this week, the graded Problem Set 10 may be collected during office hours.

Problem sets

• Problem Set 1 is due in recitation the week of September 8th--12th.
Section 1.1: 2,5,6,7,8,19, 24,42,54
Section 1.2: 3,4,10
Appendix C: 1,4,15,30, 37
• Problem Set 2 is due in recitation the week of September 15th--19th.
Section 1.5: 1, 12, 13, 15, 18, 19, 26(a),(b),(c)
Section 1.6: 5, 9, 16, 17, 20, 21, 26, 33, 36, 45, 50
• Problem Set 3 is due in recitation the week of September 22nd--26th.
Section 2.1: 5a
Section 2.2: 3, 4, 6, 8, 9, 10, 23(a).
Section 2.3: 1, 2, 4, 8, 10, 11, 12, 15, 20, 28, 32, 43.
• Problem Set 4 is due in recitation the week of October 6th--10th.
Section 2.4: 3, 4, 6, 13, 14, 16, 24, 25, 26, 30, 32, 37, 48.
Section 2.5: 2, 4, 5, 6, 7, 8, 15, 18, 20, 21, 24, 25, 27, 28, 46, 48a.
Section 2.6: 7, 8, 9, 10, 13, 20, 24.
If the usual 2nd recitation is cancelled, then Problem Set 4 is due in the 1st (and only) recitation of this week.
• Problem Set 5 is due in recitation the week of October 20th--24th.
Section 2.7: 3,5,8,13,15,17,19,20,22,35,36.
Section 2.8: 3,4,6,21,22,24,32,34.
Section 2.9: 1, 2, 4, 16, 20, 21.
Section 3.1: 1a, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 38, 44, 46, 48, 54, 56.
Section 3.2: 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 20, 29, 30, 32, 38.
• Problem Set 6 is due in recitation the week of October 27th--31st.
Section 3.4: 2, 3, 5, 6, 7, 8, 9, 10, 12, 20(a), 27, 28, 30, 35.
Section 3.5: 2, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 29, 30.
• Problem Set 7 is due in recitation the week of November 10th--14th. Because there is a midterm including the material from Chapter 3 on November 6th, the exercises from Sections 3.5 through 3.8 are for practice only, not to be written up. As always, you are expected to read through those problems and understand how to solve them.
Section 3.5: 69, 71.
Section 3.6: 3, 5, 6, 7, 9, 11, 12, 15, 18, 29, 32, 40, 43, 44.
Section 3.7: 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 19, 21, 27, 29, 30, 31, 33, 34, 35, 36.
Section 3.8: 6, 8, 15, 16, 28a.
Section 4.1: 1, 2, 3, 4, 9, 11, 12, 13, 18.
Section 4.2: 1, 2, 3, 4, 5, 6, 7, 9, 25, 29, 32, 33, 37, 41, 44.
• Problem Set 8 is due in recitation the week of November 17th--21st.
Section 4.3: 7, 9, 11, 13, 19, 23, 27, 29, 30, 31, 32, 52.
Section 4.5: 1, 2, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 45, 47, 55.
• Problem Set 9 is due in recitation the week of December 1st--5th.
Section 4.5: 3, 4, 31, 33, 35, 37, 39
Section 4.6: 10, 13, 19, 24, 31, 43
Section 4.9: 1, 3, 5, 7, 9, 11, 12, 15, 17, 19, 20, 21, 23, 25, 27, 28, 39.
• Problem Set 10 is due in recitation the week of December 8th--12th.
Section 5.1: 1,18, 19,20.
Section 5.2: 1, 17, 18, 21, 22, 25, 32, 34, 38, 40, 41, 42, 44, 49.
Section 5.3: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 24, 25, 27, 38.
Section 5.4: 7, 9, 11, 12, 13, 15.

• Text. David S. Dummitt and Richard M. Foote, Abstract algebra, 3rd ed.
• Grading policy. Midterm I: 20%, Midterm II: 20%, Final Exam: 35%, Problem Sets: 25%.

Syllabus.

• Week 1, August 31st -- September 4th
Chapter 1. Introduction to Groups, p. 16

• Week 2, September 7th -- September 11th
Chapter 2. Subgroups, p. 46
Problem Set 1 due in Thursday lecture.
• Week 3, September 14th -- September 18th
Chapter 3. Quotient Groups and Homomorphisms, p. 73
Problem Set 2 due in Thursday lecture.
• Week 4, September 21st -- September 25th
Special week because of the quiz
Chapter 4. Group actions, p. 112
DIAGNOSTIC QUIZ in Thursday lecture.
Problem Set 3 due in Thursday lecture.
• Week 5, September 28th -- October 2nd
Special week because of the holiday
Chapter 5. Direct and Semidirect Products and Abelian Groups, p. 152
No lecture on Tuesday, September 29th, the correction day.
Problem Set 4 due in Thursday lecture.
• Week 6, October 5th -- October 9th
Chapter 11. Vector Spaces, p. 408
MIDTERM 1 in Thursday lecture.
No assigned problems to be collected this week.
• Week 7, October 12th -- October 16th
Chapter 11. Vector Spaces (continued), p. 408
Problem Set 5 due in Thursday lecture.
• Week 8, October 19th -- October 23rd
Inner Product Spaces and The Spectral Theorem
Problem Set 6 due in Thursday lecture.
• Week 9, October 26th -- October 30th
Chapter 7. Introduction to Rings, p. 223
Problem Set 7 due in Thursday lecture.
• Week 10, November 2nd -- November 6th
Chapter 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains, p. 270
Problem Set 8 due in Thursday lecture.
• Week 11, November 9th -- November 13th
Chapter 9. Polynomial Rings, p. 295
Problem Set 9 due in Thursday lecture.
• Week 12, November 16th -- November 20th
Chapter 15. Commutative Rings and Algebraic Geometry, p. 656
MIDTERM 2 in Thursday lecture.
No assigned problems to be collected this week.
• Week 13, November 23rd -- November 27th
Semisimple Rings and Wedderburn's Theorem
No lecture Thursday, November 26th.

No assigned problems to be collected this week.
• Week 14, November 30th -- December 4th
Chapter 10. Introduction to Module Theory, p. 337
Problem Set 10 due in Thursday lecture.
• Week 15, December 7th -- December 11th
Chapter 12. Modules over Principal Ideal Domains, p. 456
Problem Set 11 due in Thursday lecture.

Problem Sets.

• Problem Set 2 is due in the Thursday lecture the week of Sept. 14th — 18th.
Section 2.1: 6, 10.
Section 2.2: 4, 14.
Section 2.3: 21, 26.
Section 2.4: 7, 12.
Section 2.5: 11, 16.
• Problem Set 3 is due in the Thursday lecture the week of Sept. 21st — 25th.
Section 3.1: 17, 24.
Section 3.2: 4, 12.
Section 3.3: 3, 8.
Section 3.4: 7, 8.
Section 3.5: 3, 16.
• Problem Set 4 is due in the Thursday lecture the week of Sept. 28th — Oct. 2nd.
Section 3.4: 9, 10.
Section 3.5: 6, 8.
Section 4.1: 3, 7(a) and (d).
Section 4.2: 11, 12, 13.
• Problem Set 5 is due in the Thursday lecture the week of Oct. 12th — 16th.
Section 4.3: 19, 21, 24.
Section 4.4: 2, 10.
Section 4.5: 24, 38.
Section 4.6: 3, 4.
• Problem Set 6 is due in the Thursday lecture the week of Oct. 19th — 23rd.
Section 4.6: 6, 7.
Section 5.1: 7, 8*, 9*.
Section 5.2: 8, 14.
Section 5.4: 8, 9.
* There is a small mistake in Problems 8,9 of Section 5.1. Please prove injectivity of the homomorphism from Problem 8 only for the special case occurring in Problem 9.
• Problem Set 7 is due in the Thursday lecture the week of Oct. 26th — 30th.
Section 5.5: 7, 13, 23.
Section 11.1: 6, 8, 9.
Section 11.2: 19, 26, 33.
• Problem Set 8 is due in the Thursday lecture the week of Nov. 2nd — 6th.
Section 11.3: 2, 3, 4.
Section 11.4: 4, 5.
Section 11.5: 5, 11, 13.
• Problem Set 9 is due in the Thursday lecture the week of Nov. 9th — 13th.
Problem Set on the spectral theorem ( pdf, dvi, ps).
• Problem Set 10 is due in the Thursday lecture the week of Nov. 30th — Dec. 4th.
Main Problem. Let G and G' be groups with the same finite order n. Let m be some integer divisor of n, and assume that G and G' each contain a unique normal subgroup P, resp. P', of order m. For each group, consider the associated group which is the centralizer of P, resp. P', modulo the center of P, resp. the center of P'. If G and G' are isomorphic, prove that these associated groups are also isomorphic.

Next, assume that there exists a subgroup Q of G which intersects P in only the identity element and such that P and Q generate G, i.e., G is a semidirect product of P and Q. Let f denote the induced homomorphism from Q to the outer automorphism group of P (not the usual homomorphism to the automorphism group of P). Show that the kernel of f is canonically isomorphic to the quotient of the centralizer of P by the center of P. Thus the isomorphism class of the kernel of f is an isomorphism invariant of G. In particular, observe that the hypotheses hold if we restrict to groups G and G' which are each a semidirect product of a fixed finite p-group P and a fixed group Q whose order is less than p+1. Moreover, if P is abelian then the outer automorphism group of P equals the usual automorphism group of P so that f is the usual homomorphism.

Second Problem. For every problem on Midterm 2 where you lost points, please write up a complete, correct solution to that problem (if you only lost points on a part of the problem, you can write up just the solution for that part). If you got full credit on the exam, you do not need to write up anything (you will automatically get credit for this part of the homework assignment).

• Problem Set 11 is due in the Thursday lecture the week of Dec. 7th — Dec. 11th.
This problem set is to be assigned. Section 12.1: 16, 17, 18, 19 (Please read through this sequence of exercises.)
Section 12.3: 2, 9, 13, 14, 17.

• Text. Peter J. Eccles, An Introduction to Mathematical Reasoning. Also the Geometry Notes compiled and maintained by the Stony Brook Mathematics Department.
• Grading policy. Midterm I: 25%, Midterm II: 25%, Final Exam: 30%, Problem Sets: 20%.

Syllabus.

• Week 1, August 31st -- September 4th
Chapter 1. The language of mathematics, p. 3
Chapter 2. Implications, p. 10

• Week 2, September 7th -- September 11th
Chapter 3. Proofs, p. 21
Chapter 4. Proof by contradiction, p. 30
LEC 1: No lecture Monday, Sept. 7.
Problem Set 1 due in Wed./Th. lecture.

• Week 3, September 14th -- September 18th
Chapter 5. The induction principle, p. 39
DIAGNOSTIC QUIZ in Wed./Th. lecture.
Problem Set 2 due in Wed./Th. lecture.

• Week 4, September 21st -- September 25th
Chapter 6. The language of set theory, p. 61
Chapter 7. Quantifiers, p. 74
Problem Set 3 due in Wed./Th. lecture.

• Week 5, September 28th -- October 2nd
Chapter 8. Functions, p. 89
Chapter 9. Injections, surjections and bijections, p. 101
LEC 1: Monday, Sept. 28th lecture is rescheduled to Tuesday, Sept. 29th, same hour and same room.
LEC 2: No lecture Tues., Sept. 29th.
Problem Set 4 due in Wed./Th. lecture.

• Week 6, October 5th -- October 9th
Chapter 10. Counting, p. 123
MIDTERM 1 in Wed./Th. lecture.
No assigned problems to be collected this week.

• Week 7, October 12th -- October 16th
Chapter 11. Properties of finite sets, p. 133
Problem Set 5 due in Wed./Th. lecture.

• Week 8, October 19th -- October 23rd
Chapter 12. Counting functions and subsets, p. 144
Chapter 13. Number systems, p. 157
Problem Set 6 due in Wed./Th. lecture.

• Week 9, October 26th -- October 30th
Chapter 14. Counting infinite sets, p. 170
Problem Set 7 due in Wed./Th. lecture.

• Week 10, November 2nd -- November 6th
Chapter 19. Congruence of integers, p. 229
Chapter 21, Congruence classes and the arithmetic of remainders, p. 250
Problem Set 8 due in Wed./Th. lecture.

• Week 11, November 9th -- November 13th
Chapter 21. Congruence classes (continued), p. 250
Chapter 22. Partitions and equivalence relations, p. 262
Problem Set 9 due in Wed./Th. lecture.

• Week 12, November 16th -- November 20th
Geometry Notes §1 (Introduction) — §4(Protractor Axiom)
MIDTERM 2 in Wed./Th. lecture.
No assigned problems to be collected this week.

• Week 13, November 23rd -- November 27th
Geometry Notes §5 (Triangles)
LEC 1: No lecture Wednesday, November 25th.
LEC 2: No lecture Thursday, November 26th.

No assigned problems to be collected this week.

• Week 14, November 30th -- December 4th
Geometry Notes §6 (Parallels)
Problem Set 10 due in Wed./Th. lecture.

• Week 15, December 7th -- December 11th
Geometry Notes §7 (Similarity)
Problem Set 11 due in Wed./Th. lecture.

Problem Sets.

• Problem Set 1 is due in the Wed./Th. lecture the week of Sept. 7th — 11th.
Chapter 1, pp. 8 — 9: 1.1, 1.2, 1.3, 1.4, 1.5.
Chapter 2, pp. 19 — 20: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6
Problems I, pp. 53 — 54: 1, 2, 5
• Problem Set 2 is due in the Wed./Th. lecture the week of Sept. 14th — 18th.
Chapter 3, p. 26: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8.
Chapter 4, pp. 37 — 38: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7.
Problems I, pp. 53 — 54: 3, 4, 6, 7, 8, 9, 10, 11.
• Problem Set 3 is due in the Wed./Th. lecture the week of Sept. 21st — 25th.
Chapter 5, pp. 51 — 52: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7.
Problems I, pp. 54 — 57: 12, 13, 16, 17, 18, 21, 25.
• Problem Set 4 is due in the Wed./Th. lecture the week of Sept. 28th — Oct. 2nd.
Problems I, pp. 55 — 56: 14, 19.
Chapter 6, pp. 72 — 73: 6.1, 6.2, 6.3, 6.4, 6.5.
Problems II, pp. 115 — 116: 1, 3, 4, 6.
• Problem Set 5 is due in the Wed./Th. lecture the week of Oct. 12th — 16th.
Chapter 7, pp. 86— 88: 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9.
Chapter 8, pp. 99— 100: 8.1, 8.2, 8.3, 8.4, 8.5.
Chapter 9, pp. 113— 114: 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7.
Problems II, pp. 117— 118: 13, 14, 15, 16, 18, 19.
In Exercise 19, you are free to assume that X and Y are each the set Z of integers. This does not change the main part of the problem, but it does simplify one technical aspect.
• Problem Set 6 is due in the Wed./Th. lecture the week of Oct. 19th — 23rd.
Chapter 10, p. 132: 10.1, 10.2, 10.3, 10.4.
Chapter 11, p. 143: 11.1, 11.2, 11.3, 11.4, 11.5, 11.6.
Problems III, pp. 182 — 184: 1, 3, 5, 6, 10, 11, 14.
• Problem Set 7 is due in the Wed./Th. lecture the week of Oct. 26th — 30th.
Chapter 12, p. 155 — 156: 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7.
Problems III, pp. 184 — 185: 16, 17, 18.
• Problem Set 8 is due in the Wed./Th. lecture the week of Nov. 2nd — 6th.
Chapter 13, p. 169: 13.1, 13.2, 13.3, 13.4, 13.5.
Chapter 14, p. 181: 14.1, 14.2, 14.3, 14.4.
Problems III, p. 186: 23, 24, 25, 26.
• Problem Set 9 is due in the Wed./Th. lecture the week of Nov. 9th — 13th.
Chapter 19, p. 239: 19.1, 19.2, 19.3, 19.4, 19.5.
Chapter 21, p. 261: 21.1, 21.2, 21.3, 21.4, 21.5, 21.6.
Problems V. pp. 271 — 273: 1, 3, 7, 13, 17, 18.
• Problem Set 10 is due in the Wed./Th. lecture the week of Nov. 30th — Dec. 4th.
Geometry Notes: 2.3, 2.4, 2.5, 2.6.
Geometry Notes: 3.1, 3.2, 3.3, 4.4, 4.7, 4.8.
• Problem Set 11 is due in the Wed./Th. lecture the week of Dec. 7th — Dec. 11th.
Geometry Notes: 5.1, 5.2, 5.3, 5.4, 5.6, 5.7, 5.9, 5.10.
Geometry Notes: 5.5, 5.8, 5.11.

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Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631-632-8270
Fax: 631-632-7631
Jason Starr