## 01:640:350:04 Linear Algebra Section

04 | 11407 | Woodward, Christopher | Lecture | TF2 | 1020 A - 1140 | BE-250 | LIV |

## This course is a proof-based continuation of Math 250, covering Abstract vector spaces and linear transformations, inner product spaces, diagonalization, and canonical forms.

*Prerequisites: *

- CALC4, Math 250 and Math 300

Text: *Linear Algebra* (4th ed.), by Friedberg, Insel and Spence,

Prentice Hall, 2003 ISBN 0-13-008451-4. For this section 04, any recent edition of the textbook should be sufficient.

- Clas
TF2 1020 A - 1140 BE-250 LIV - Office Hours: Wednesday 2-3pm, Hill 726
- Contact Information: e-mail ctw@math.rutgers.edu

The course is strongly based on Math 250. However, we'll work axiomatically, starting from the abstract notions of vector space and of linear transformation. Much of the homework and many of the exam and quiz problems will require you to write precise proofs, building on your proof-writing experience in Math 300. From this more abstract viewpoint, we'll be developing linear algebra far beyond Math 250, with new insight and new applications.

Class attendance is very important. A lot of what we do in class will involve collective participation. We will cover the topics indicated in the syllabus below, but the dates that we cover some of the topics might be adjusted during the semester, depending on class discussion, etc. Such adjustments, along with the almost-weekly homework assignments, will be announced in class and also posted on this webpage, so be sure to check this webpage regularly. Absences from a single class due to minor illnesses should be self-reported using the university system; for longer absences, students should email me with the situation. I reserve the right to lower the course grade up to one full letter grade for poor attendance.

Make-ups for exams are generally not given; if a student has an extremely good reason (e.g. documented medical emergency) I may re-arrange the grading scheme to accomodate.

Problem sets are due on most Tuesdays. There are no problems due on the two midterm-exam Tuesdays.

Note that we will cover significant material from all the chapters in the book, Chapters 1-7, but we will cover Chapter 7 before Chapter 6. This is because the material in Chapter 7 is a natural continuation of the material in Chapter 5 on the theory of eigenvalues, eigenvectors and diagonalizability. Chapter 6 also concerns eigenvalues, eigenvectors and diagonalizability, but this time, based on a generalization of the theory of dot products.

Quizzes may be given at the ends of a few class sessions. The dates of these quizzes, and the topics covered, will be announced in advance.

Grading policy: First midterm exam: 100 points; Second midterm exam: 100 points; Problem sets and quizzes: 100 points; Final exam: 200 points (Total: 500 points).

#### Tentative Course Syllabus

Week | Lecture dates | Sections | topics |
---|---|---|---|

1 | 9/4 (T) | Chapter 1 | Abstract vector spaces & subspaces |

2 | 9/7, 9/11 | Chapter 1 | Span of subsets, linear independence |

3 | 9/14, 9/18 | Chapter 1 | Bases and dimension |

4 | 9/21, 9/25 | Chapter 2 | Linear transformations |

5 | 9/28, 10/2 | Chapter 2 | Change of basis, dual spaces |

6 | 10/5, 10/9 | Ch. 1-2 | Review and Exam 1 (10/9) |

7 | 10/12, 10/16 | Chapter 3 | Rank and Systems of Linear Equations |

8 | 10/19, 10/23 | Chapter 4 | Determinants and their properties |

9 | 10/26, 10/30 | Chapter 5 | Eigenvalues/eigenvectors |

10 | 11/2, 11/6 | Chapter 5 | Diagonalization, Markov Chains |

11 | 11/9, 11/13 | Chapter 6 | Inner Product spaces |

12 | 11/16 | Chapter 6 | Unitary and Orthogonal operators |

13 | 11/21, 11/27 | Ch.3,4,5,7 | Review and Exam 2 (11/27) |

14 | 11/30, 12/4 | Chapter 7 | Orthogonal diagonalization |

15 | 12/7, 12/11 | Chapter 7 | Jordan canonical form |

17 | 12/21 (Fri) | 8-11am | Final Exam Location TBA |

#### Exam Dates

The exam dates are listed in the schedule above. Any conflict (such as with a religious holiday) should be reported to me at the beginning of the semester, so that the exam may be re-scheduled.

### Special Accommodations

Students with disabilities requesting accommodations must follow the procedures outlined at https://ods.rutgers.edu/students/applying-for-services

### Academic Integrity

All Rutgers students are expected to be familiar with and abide by the academic integrity policy. Violations of the policy are taken very seriously. In particular, your work should be your own; you are responsible for properly crediting all help with the solution.

### Problem Sets

The Problem Sets are available in the assignments directory on the course Sakai site.

Problem sets should be hand-written in reasonably clear writing, with an explanation of any assistance given. Type-written assignments are allowable only by special arrangement (disability etc.) Scans of problem sets may be submitted electronically in emergencies (illness or accident) by upload to Sakai.

Some basic writing guidelines are as follows. Please write in complete sentences; avoid starting each sentence with a symbol; ensure that each variable or notation is defined; number sentences or formulas as necessary so that you may refer back to them. To prove a "for all x", usually begin with "Let x be a ...". To prove an "there exists x" statement, you must construct a particular x satisfying the given property, so "Define x to be ...". To prove a that property A implies property B, begin with "Assume Property A...." Then deduce Property B. Sets are equal if they have the same elements; functions are equal if they have the same values; to prove something does not satisfy a list of axioms; it suffices to show that one of the axioms fails. On both problem sets and exams you may use properties in the text or class (referring to them by page or date) if they come before the problem you are solving in the development of the material.

#### Problem Sets from 2017

Problems in pdf. Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Practice problems in pdf for the first midterm. Last year's exam with Answers.

Problem in pdf.)

(Problem in pdf)

(Problems in pdf)

Problems in pdf)

Problems in pdf.

Answers and practice problems for the final.

Pratice problems for the second exam:

(Problems in pdf)

#### Recommended Practice Problems (the problem sets from 2016)

Sept. 13 | 1.2 #17; 1.3 #19,23; 1.4 #11,13; 1.5 #9,15 |

Sept. 20 | 1.6 # 20,21,26,29; 1.7 #5,6 |

Sept. 27 | 2.1 #3,11,28; 2.2 #4; 2.3 #12; 2.4 #15,17 |

October 4 | 2.5 #3(d),7(a,b),13; 2.6 #5,10; Show that F[x]* ≅ F[[x]]. |

October 18 | 3.1 #6,12; 3.2 #5(b,d,h),17; 3.3 #8,10; 3.4 #8,15 If an nxn matrix A has each row sum 0, some Ax=b has no solution. |

October 25 | 4.1 #10(a,c); 4.2 #23; 4.3 #12,22(c),25(c); 4.4 #6; 4.5 #11,12 |

Nov. 1 | 5.1 #3(b),20,21; 5.2 #4,9(a),12; Show that the cross product induces an isomorphism between R³ and Λ²(R³). |

Nov. 8 | 5.2 #18(a),21; 5.3 #2(d,f); 5.4 #6(a),13,19,25 |

Nov. 15 | 7.1 #3(b),9(a),13; 7.2 #3,14,19(a); 7.3 #13,14; Find all 4x4 Jordan canonical forms of T satisfying T²=T³. |

Dec. 13 | 6.1; #6,11,12,17; 6.2 #2a,6,11; 6.8 #4(a,c,d),11 |