## Math 110: Linear Algebra (Fall 2010)

Description: Vector spaces. Linear transformations and matrices. Linear systems of equations. Determinants. Eigenvalues and eigenvectors. Inner product spaces. Self-adjoint, normal, and unitary maps. Course control number: 54239. Per-Olof Persson, persson@berkeley.edu, 1089 Evans, Phone (510) 642-6947 Office hours: Tue 9:30am - 11:00am and Fri 2:00pm - 3:30pm in 1089 Evans MWF 1-2pm, Room 100 Lewis S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 4th edition, Prentice Hall, 2002 (Amazon). Lecture Slides, Chapter 1 - Vector Spaces (Full page, 6 per page) Lecture Slides, Chapter 2 - Linear Transformations and Matrices (Full page, 6 per page) Lecture Slides, Chapter 3 - Elementary Matrix Operations and Systems of Linear Equations (Full page, 6 per page) Lecture Slides, Chapter 4 - Determinants (Full page, 6 per page) Lecture Slides, Chapter 5 - Diagonalization (Full page, 6 per page) Lecture Slides, Chapter 6 - Inner Product Spaces (Full page, 6 per page) Notes on sets, logic, and mathematical language by Prof. George Bergman Solutions, problem 2.6.13 (a-b), from in-class office hours

Announcements: (Dec 15): Final Exam Results Here is the Final Exam and the solutions. The average score was 64.6 out of 100, the standard deviation 24.1, and the median 68. The approximate grade levels are: A) 82 - 100, B) 63 - 81, C) 41 - 62, D) 20 - 40, F) 0 - 19 (Dec 1): Final Exam The Final Exam will be held in the regular classroom 100 Lewis on Wednesday December 15 between 7:00pm - 10:00pm. Make sure you arrive on time. The exam will cover everything we have done, namely, sections 1.2-1.6, 2.1-2.6, 3.1-3.4, 4.1-4.4, 5.1-5.4, 6.1-6.5 of the textbook and homeworks 1-11. Bring pens/pencils/eraser and one 2-sided sheet of notes (letter size). No calculator, no book, and no other notes are allowed. DSP Students: Please come to room 736 Evans at 5:30pm, and a GSI will arrange for you to take the exam. Here are some sample final exams from previous Math 110 courses. Note: 1) These might have covered slightly different material. 2) Different instructors write different exams, so do not draw any conclusions about the exact format of our midterm exam based on these exams. 3) The solutions should be considered "solution sketches", that is, they might not have been given full credit if we graded the exams. (Oct 5): Midterm 2 Results Here are Midterm 2 and the solutions. The average score was 22.4 out of 40, the standard deviation 9.4, and the median 23. The approximate grade levels are: A) 30 - 40, B) 22 - 29, C) 14 - 21, D) 7 - 13, F) 0 - 6 (Oct 28): Midterm 2 Midterm 2 will be held in the regular classroom 100 Lewis on Monday November 8 between 1:10pm - 2:00pm. Make sure you arrive on time. The exam will cover everything we have done, namely, sections 1.2-1.6, 2.1-2.6, 3.1-3.4, 4.1-4.4, 5.1-5.2 of the textbook and homeworks 1-8. Bring pens/pencils/eraser and one 2-sided sheet of notes (letter size). No calculator, no book, and no other notes are allowed. DSP Students: Please come to room 939 Evans at 1:00pm, and a GSI will arrange for you to take the exam. Here are some sample second midterms from previous Math 110 courses. Note: 1) These might have covered slightly different material, and might have been 80 minutes long instead of 50 minutes. 2) Different instructors write different exams, so do not draw any conclusions about the exact format of our midterm exam based on these exams. 3) The solutions should be considered "solution sketches", that is, they might not have been given full credit if we graded the exams. (Oct 5): Midterm 1 Results Here are Midterm 1 and the solutions. The average score was 19.1 out of 40, the standard deviation 9.1, and the median 18. The approximate grade levels are: A) 28 - 40, B) 20 - 27, C) 13 - 19, D) 6 - 12, F) 0 - 5 (Sep 26): Midterm 1 Midterm 1 will be held in the regular classroom 100 Lewis on Monday October 4 between 1:10pm - 2:00pm. Make sure you arrive on time. The exam will cover everything we have done, namely, sections 1.2-1.6, 2.1-2.5 of the textbook and homeworks 1-4. Bring pens/pencils/eraser and one 2-sided sheet of notes (letter size). No calculator, no book, and no other notes are allowed. DSP Students: Please come to room 939 Evans at 1:00pm, and a GSI will arrange for you to take the exam. Here are some sample first midterms from previous Math 110 courses. Note: 1) These might have covered slightly different material, and might have been 80 minutes long instead of 50 minutes. 2) Different instructors write different exams, so do not draw any conclusions about the exact format of our midterm exam based on these exams. 3) The solutions should be considered "solution sketches", that is, they might not have been given full credit if we graded the exams.

Homework:
HW Date Problems Solutions
01 Wed 09/08 1.2: 1, 8, 10, 16, 19
1.3: 1, 3, 8ce, 13, 23, 30
1.4: 1, 2bf, 5aeg, 12, 15
PDF
02 Wed 09/15 1.5: 1, 2ef, 13, 17, 18
1.6: 1, 2abc, 3de, 14, 17, 22, 26, 31, 33
PDF
03 Wed 09/22 2.1: 1, 9, 11, 15, 18, 20, 26, 35
2.2: 1, 3, 10, 12, 13
PDF
04 Wed 09/29 2.3: 1, 3, 11, 13, 17
2.4: 1, 2, 6, 7, 9, 17
2.5: 1, 3ab, 6ab, 8, 10
PDF
05 Wed 10/13 2.6: 1, 4, 7, 14, 15
3.1: 1, 3, 5
PDF
06 Wed 10/20 3.2: 6, 14
3.3: 1, 2f, 3f
3.4: 1, 2ef, 10
4.1: 1
PDF
07 Wed 10/27 4.1: 9, 10
4.2: 1, 3, 12, 20, 23, 25, 26
4.3: 1, 10, 11, 14, 15, 23
PDF
08 Wed 11/03 4.4: 5, 6
5.1: 1, 3, 7, 9, 14, 17, 22
5.2: 1, 7, 8, 12, 13
PDF
09 Wed 11/17 5.3: 1, 2bdegi, 3, 4
5.4: 1, 3, 4, 5, 13, 17, 18
PDF
10 Wed 11/24 6.1: 1, 3, 9, 11, 17, 20
6.2: 1, 2abi, 13, 15
6.3: 1, 3, 12, 14
PDF
11 Wed 12/01 6.4: 1, 2, 12, 18
6.5: 1, 2, 15
PDF
Syllabus:
Lec Date Topic Other
01 Fri 08/27 1.2: Vector Spaces
02 Mon 08/30 1.3: Subspaces
03 Wed 09/01 1.4: Linear Combinations and Systems of Linear Equations
04 Fri 09/03 1.5: Linear Dependence and Linear Independence
Mon 09/06 Labor Day - No Lecture
05 Wed 09/08 1.6: Bases and Dimension HW1 Due
06 Fri 09/10 1.6: (cont'd)
07 Mon 09/13 2.1: Linear Transformations, Null Spaces, and Ranges
08 Wed 09/15 2.1: (cont'd) HW2 Due
09 Fri 09/17 2.2: The Matrix Representation of a Linear Transformation
10 Mon 09/20 2.3: Composition of Linear Transformations and Matrix Multiplication
11 Wed 09/22 2.4: Invertibility and Isomorphisms HW3 Due
12 Fri 09/24 2.4: (cont'd)
13 Mon 09/27 2.5: The Change of Coordinate Matrix
14 Wed 09/29 2.6: Dual Spaces HW4 Due
15 Fri 10/01 Review
16 Mon 10/04 Midterm 1 - In class, 100 Lewis, 1-2pm Midterm 1
17 Wed 10/06 3.1: Elementary Matrix Operations and Elementary Matrices
18 Fri 10/08 3.2: The Rank of a Matrix and Matrix Inverses
19 Mon 10/11 3.2: (cont'd)
20 Wed 10/13 3.3: Systems of Linear Equations - Theoretical Aspects HW5 Due
21 Fri 10/15 3.4: Systems of Linear Equations - Computational Aspects
22 Mon 10/18 4.1: Determinants of Order 2
23 Wed 10/20 4.2: Determinants of Order n HW6 Due
24 Fri 10/22 4.2: (cont'd)
25 Mon 10/25 4.3: Properties of Determinants
26 Wed 10/27 4.4: Summary - Important Facts about Determinants HW7 Due
27 Fri 10/29 5.1: Eigenvalues and Eigenvectors
28 Mon 11/01 5.2: Diagonalizability
29 Wed 11/03 5.3: Matrix Limits and Markov Chains HW8 Due
30 Fri 11/05 Review
31 Mon 11/08 Midterm 2 - In class, 100 Lewis, 1-2pm Midterm 2
32 Wed 11/10 5.4: Invariant Subspaces and the Cayley-Hamilton Theorem
33 Fri 11/12 5.4: (cont'd)
34 Mon 11/15 6.1: Inner Products and Norms
35 Wed 11/17 6.2: The Gram-Schmidt Orthogonalization Process and Orthogonal Complements HW9 Due
36 Fri 11/19 6.3: The Adjoint of a Linear Operator
37 Mon 11/22 6.4: Normal and Self-Adjoint Operators
38 Wed 11/24 6.4: (cont'd) HW10 Due
Fri 11/26 Thanksgiving - No lecture
39 Mon 11/29 6.5: Unitary and Orthogonal Operators and Their Matrices
40 Wed 12/01 Review HW11 Due
41 Fri 12/03 Review
Reading/Review/Recitation Week 12/6-12/10 - No lectures
Wed 12/15 Final Exam - 7-10pm Final Exam
GSIs and Discussion Sections:
Sec Time Room GSI E-mail Office Office hours
01 Wed 8am - 9am 87 Evans D. Penneys dpenneys@math.berkeley.edu 1049 Evans Tue 10-11am and 5-6pm, Wed 9-10am
02 Wed 9am - 10am 2032 Valley LSB C. Mitchell cmitch5@math.berkeley.edu 745 Evans Mon 2-4pm, Tue 1-2pm
03 Wed 10am - 11am B51 Hildebrand D. Beraldo beraldo@math.berkeley.edu 835 Evans Tue 4-6pm, Thu 4:30-5:30pm
04 Wed 11am - 12pm B51 Hildebrand D. Beraldo beraldo@math.berkeley.edu 835 Evans Tue 4-6pm, Thu 4:30-5:30pm
05 Wed 12pm - 1pm 75 Evans C. Mitchell cmitch5@math.berkeley.edu 745 Evans Mon 2-4pm, Tue 1-2pm
07 Wed 2pm - 3pm 87 Evans C. Mitchell cmitch5@math.berkeley.edu 745 Evans Mon 2-4pm, Tue 1-2pm
08 Wed 9am - 10am 3113 Etcheverry I. Ventura iventura@math.berkeley.edu 1039 Evans Tue 11:30am-1:30pm, Thu 4-6pm
09 Wed 2pm - 3pm 3 Evans D. Penneys dpenneys@math.berkeley.edu 1049 Evans Tue 10-11am and 5-6pm, Wed 9-10am
10 Wed 12pm - 1pm 310 Hearst I. Ventura iventura@math.berkeley.edu 1039 Evans Tue 11:30am-1:30pm, Thu 4-6pm

Grading and policies: Homework: Weekly homework is posted on the course web page, and it is due in each Wednesday discussion section (except for the first week and weeks with midterm exams). Collaboration on the homework is encouraged, but each student must write his/her own solutions and not copy them from anyone else. Only some of the problems from each homework will be graded. Late homework will not be accepted, but the two lowest scores will be dropped when computing the grade. Exams: There will be two in-class midterm exams, scheduled for Monday October 4 and Monday November 8 between 1pm - 2pm. The final exam will be given on Wednesday December 15 between 7pm - 10pm. You may bring one (ordinary sized) sheet of paper with writing on both sides to the exams. Apart from this one sheet, the exams are "closed book". In particular, you may not bring textbooks, notebooks, or calculators. If there is an emergency alarm during the midterms or the final exam, leave the exam at the desk and walk out. You may of may not be allowed back to complete the work. Grade corrections: The grades for the exams will be changed only if there is a clear error on the part of the grader, such as adding up marks incorrectly. Problems must be brought to the attention of the GSI immediately after the exams are returned. Grades: The final grade will be based on weekly homework assignments (25%), Midterm 1 (15%), Midterm 2 (15%), and the Final Exam (45%). The lowest midterm grade can be dropped and replaced by the final exam grade. This allows you to miss one midterm, but your chances are improved if you take both. If you miss both midterm exams or if you miss the Final Exam, you will fail the course. Incomplete grades: Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or genuine personal/family emergency. Falling behind in this course or problems with workload in other courses are not acceptable reasons. Special arrangements: If you are a student with a disability registered by the Disabled Student Services (DSS) on UCB campus and if you require special arrangements during exams, you must provide the DSS document and make arrangements via email or office hours at least 10 days prior to each exam, explaining your circumstances and what special arrangements need to be done. Also see your GSI as soon as possible if you need special arrangements during the sections or for submitting the homeworks.