# Mth 221 Assignment 1 Solution

General information | Course description | Assignments | Homework schedule | Grading | Links | Fine print

## General information

**Goal:** Students will become proficient in both using and understanding the theory and algorithms of linear algebra and will learn how to write rigorous mathematical arguments.

**Lectures:** Wednesday and Friday, 13:25 – 14:40, Physics Building 119

**Text:***Linear Algebra: A Geometric Approach*, by Ted Shifrin and Malcolm Adams, second edition.

### Contact information for the Instructor

**Name:** Professor Ezra Miller **Address:** Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320 **Office:** Physics 209 **Phone:** (919) 660-2846 **Email:***ezramath.duke.edu***Webpage:***https://math.duke.edu/people/ezra-miller***Course webpage:**you're already looking at it... but it's *https://services.math.duke.edu/~ezra/221/221.html***Office hours:** Tuesday 14:00 – 15:00 & Wednesday 12:00 – 13:15, in Physics 209

### Help Room

Mondays and Thursdays, 7:00 - 10:00 in Carr 137, from September 4^{th}to December 11

^{th}.

See here for more information.

## Course description

**Goal:** Students will become proficient in both using and understanding the theory and algorithms of linear algebra, and they will learn how to write rigorous mathematical arguments.

**Course content:** Chapters 1 – 6 of the course text, by Shifrin & Adams

Most items constitute one lecture each; some fill two lectures:

- vectors
- -dimensional geometry
- Gaussian elimination
- linear systems
- matrix algebra
- linear transformations
- elementary matrices; transpose
- linear subspaces
- linear independence
- bases; dimension
- abstract vector spaces
- inner products
- projections; least squares
- orthonormal bases; Gram-Schmidt algorithm
- changes of basis
- abstract linear transformations
- determinants
- formulas for determinants
- eigenvalues and eigenvectors
- Markov processes
- spectral theorem

**Prerequisite:**Second-semester calculus (Math 122, 112L, or 122L)

## Assignments

- Weekly homework will be collected and graded.
- There will be two in-class exams:
- Thursday 6 October
- Thursday 17 November

- There will be a final exam: Saturday 16 December, 19:00–22:00 in Physics 119
- Quizzes may be given if homework performance lags.

**Policies regarding graded work**

- Tentative due dates for the homework assignments this semester are listed in the table below.
- You should work on the homework for a section immediately after the class in which it is covered.
- Homework for the sections covered in a week will be due and collected at the beginning of the following Tuesday's class.
- Late homework will not be accepted.
- Missed quizzes or exams: in general there will be no make-ups, but some accommodation may be possible in one of the following four situations: personal emergencies or tragedies, an incapacitating illness, a religious holiday, or varsity athletic participation. Please visit these web-pages now to familiarize yourself with the procedures.
- Collaboration on homework is encouraged while you discuss the search for solutions, but when it comes time to write them down, the work you turn in must be yours alone: you are not allowed to consult anyone else's written solution, and you are not allowed to share your written solutions. (It is very easy to tell when solutions have been copied or written together.) If you collaborate, you must indicate—on the homework page—who your collaborators were.
- Collaboration of any sort on quizzes, in-class exams, and the final exam is not permitted: you must work completely independently without giving or receiving help from others.
- Students are expected to adhere to the Duke Community Standard. You must reaffirm your committment to these standards on all work.
- If a student is responsible for academic dishonesty on a graded item in this course, then the student will have an opportunity to admit the infraction and, if approved by the Office of Student Conduct, resolve it directly through a faculty-student resolution agreement; the terms of that agreement would then dictate the grading response to the assignment at issue. If the student is found responsible through the Office of Student Conduct and the infraction is not resolved by a faculty-student resolution agreement, then the student will receive a score of zero for that assignment, and the instructor reserves the right to further reduce the final grade for the course by one or more letter grades—possibly to a failing grade—at the discretion of the instructor.
- Computer policy: You may use a computational aid for the homework but I recommend avoiding it as much as possible, particularly since electronic devices of all sorts are not allowed on quizzes, in-class exams, and the final exam. Portable electronic devices that are visible or audible during exams will be confiscated until the exam period ends.
- All submitted work (including quizzes and exams, in addition to homework) must be written neatly and legibly. Instead of erasing please use a single line crossout.
- If a question requires more than a single expression or equation, your response must be phrased in complete sentences.
- The logic of a proof must be completely clear for full credit.
**Staple**multiple-page submissions or they will be returned ungraded. Paper clips do not suffice; pages can and often do become separated.

## Homework schedule

If a lecture or assignment hasn't been posted, and you think it should have been, then please do email me. Sometimes I encounter problems (such nas, for example, the department's servers going down) while posting assignments; other times, I might simply have forgotten to copy the updated files into the appropriate directory, or to set the permissions properly.

Read and study the text carefully before attempting the assignments. Make sure you fully understand the given proofs and examples; note that there are examples in the text similar to most of the homework problems. The material in gray shaded boxes consists of definitions; learn them precisely. (Often when students say they do not know how to do a problem it is because they don't know the definitions of the terms in the problem.) The material in blue shaded boxes introduces points of logic and techniques of proof that you will find helpful in writing your arguments. If you have trouble understanding something in the text after working on it for a while, then see me in office hours or e-mail me.

# | Date | Sections | Topic | Assignment due |
---|---|---|---|---|

1. | Wed 30 Aug | 1.1 – 1.2 | vectors | |

2. | Fri 1 Sep | 1.2 – 1.3 | -dimensional geometry | |

3. | Wed 6 Sep | 1.4 | matrix multiplication | 1.1: 6(a,c,g), 7, 8, 9, 21, 22, 23, 25, 29 1.2: 1(b,d,g), 2(b,d,g), 4, 9, 11, 13 (no geometric interpretation necessary), 16, 18 |

4. | Fri 8 Sep | 1.4 | Gaussian elimination | |

5. | Wed 13 Sep | 1.5 | linear systems | 1.3: 1(a,c,f), 3(a,d,e), 5, 8, 10, 12 1.4: 1, 3(a–f), 4(d,f), 10, 11, 12, 13, 15 |

6. | Fri 15 Sep | 1.5, 1.6.1 | solving linear systems | |

7. | Wed 20 Sep | 2.2 – 2.3 | linear transformations | 1.5: 1, 2(a, b), 3(a, c), 4a, 6, 10, 12, 13, 14 1.6: 5, 7, 9, 11 |

8. | Fri 22 Sep | 2.1 – 2.2 | matrix algebra | |

9. | Wed 27 Sep | 2.4 – 2.5 | elementary matrices transpose | 2.1: 1(a, c, f), 2, 5, 6, 7, 8, 12(a, b, d), 14 2.2: 5, 7, 8 2.3: 1(b, d, f), 2(a, c, d), 4, 8, 11, 13, 16 |

10. | Fri 29 Sep | 3.1 – 3.2 | linear subspaces | |

11. | Wed 4 Oct | FIRST MIDTERM EXAM | 2.4: 7, 12 2.5: 1(a,f,j), 4, 8, 9, 12, 15, 19(a,b), 22, 23 | |

12. | Fri 6 Oct | 3.2 | linear subspaces | |

13. | Wed 11 Oct | 3.3 | linear independence | 3.1: 1, 2(a,c,d), 6, 9(b,c), 10, 12, 13, 14 |

14. | Fri 13 Oct | 3.3 – 3.4 | bases; dimension | |

15. | Wed 18 Oct | 3.4 | bases; dimension | 3.2: 1, 2(a,b), 10, 11 3.3 1, 2, 8, 10, 11, 14, 15, 19, 21, 22 |

16. | Fri 20 Oct | 3.6 | abstract vector spaces | |

17. | Wed 25 Oct | 3.6 – 4.1 | inner products; projections | 3.3: 5(a,b) 3.4: 3(a, b, d), 4, 8, 17, 20, 24 |

18. | Fri 27 Oct | 4.1 – 4.2 | least squares; orthonormal bases; Gram-Schmidt | |

19. | Wed 1 Nov | 4.3 | change of basis | 3.6: 1, 2(a, c, d), 3(a, c, f), 4, 6(a, b), 9, 13, 14(b, c), 15(a, b) |

20. | Fri 3 Nov | 4.4 | abstract linear transformations | |

21. | Wed 8 Nov | 4.3 – 4.4 | review of change of basis | 4.1: 1(a, b), 3, 6, 7, 9, 11, 13, 15 4.2: 2(b,c), 3, 6, 7(a, b), 8a, 9a, 11, 12(a, b) |

22. | Fri 10 Nov | 5.1 | determinants | |

23. | Wed 15 Nov | 5.2 | formulas for determinants | 4.3: 3, 7, 9, 12, 18, 19, 20, 21 4.4: 2, 5, 7, 8, 11, 13, 14 |

24. | Fri 17 Nov | SECOND MIDTERM EXAM | SECOND MIDTERM EXAM | |

Wed 22 Nov | no class: Thanksgiving | |||

Fri 24 Nov | no class: Thanksgiving | |||

25. | Wed 29 Nov | 6.1 | eigenvalues and eigenvectors | 5.1: 1(a, b, c), 2, 3, 4, 7, 9(a), 10, 11 5.2: 1a, 3, 4, 5(a,c,f), 7, 8, 10 |

26. | Fri 1 Dec | 6.2 | diagonalizability | |

27. | Wed 6 Dec | 7.1; 6.4 | Jordan form; spectral theorem | 6.1: 1 (do as many as you can stand!), 2, 3, 4, 6, 10, 12, 14 6.2: 1 (do as many as you can stand!), 3, 4, 6, 11, 16(a-c) |

28. | Fri 8 Dec | 7.3 | matrix exponentials; systems of ODE | |

Sat 16 Dec | FINAL EXAM, 19:00 – 22:00 | *optional* 6.4: 1, 2, 3, 4, 5, 8, 10, 11, 13 *optional* 7.1: 4, 6, 7, 8, 14, 16 *optional* 7.3: 1, 4, 5, 8, 9, 10, 13, 14 |

## Grading scheme

**Final course grades**:

- 15% Homework
- 25% Midterm #1
- 25% Midterm #2
- 35% Final exam

## Links

**University academic links**

**Departmental links**

### The fine print

I will do my best to keep this web page for Math 221 current, but this web page is not intended to be a substitute for attendance. Students are held responsible for all announcements and all course content delivered in class.Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

*ezramath.duke.edu**Fri Dec 8 03:16:23 EST 2017*

Back to personal homepage Ezra's Math homepage Duke Math homepage

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