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 4th to December 11th.
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.

#DateSectionsTopicAssignment due
  1.Wed 30 Aug1.1 – 1.2vectors 
  2.   Fri   1 Sep1.2 – 1.3-dimensional geometry 
  3.Wed   6 Sep1.4matrix multiplication1.1: 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 Sep1.4Gaussian elimination 
  5.Wed 13 Sep1.5linear systems1.3: 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 Sep1.5, 1.6.1solving linear systems 
  7.Wed 20 Sep2.2 – 2.3linear transformations1.5: 1, 2(a, b), 3(a, c), 4a, 6, 10, 12, 13, 14
1.6: 5, 7, 9, 11
  8.   Fri 22 Sep2.1 – 2.2matrix algebra 
  9.Wed 27 Sep2.4 – 2.5elementary 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 Sep3.1 – 3.2linear subspaces 
11.Wed   4 Oct FIRST MIDTERM EXAM2.4: 7, 12
2.5: 1(a,f,j), 4, 8, 9, 12, 15, 19(a,b), 22, 23
12.   Fri   6 Oct3.2linear subspaces 
13.Wed 11 Oct3.3linear independence3.1: 1, 2(a,c,d), 6, 9(b,c), 10, 12, 13, 14
14.   Fri 13 Oct3.3 – 3.4bases; dimension 
15.Wed 18 Oct3.4bases; dimension3.2: 1, 2(a,b), 10, 11
3.3 1, 2, 8, 10, 11, 14, 15, 19, 21, 22
16.   Fri 20 Oct3.6abstract vector spaces 
17.Wed 25 Oct3.6 – 4.1inner products; projections3.3: 5(a,b)
3.4: 3(a, b, d), 4, 8, 17, 20, 24
18.   Fri 27 Oct4.1 – 4.2least squares; orthonormal bases; Gram-Schmidt 
19.Wed  1 Nov4.3change of basis3.6: 1, 2(a, c, d), 3(a, c, f), 4, 6(a, b), 9, 13, 14(b, c), 15(a, b)
20.   Fri  3 Nov4.4abstract linear transformations 
21.Wed  8 Nov4.3 – 4.4review of change of basis4.1: 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 Nov5.1determinants 
23.Wed 15 Nov5.2formulas for determinants4.3: 3, 7, 9, 12, 18, 19, 20, 21
4.4: 2, 5, 7, 8, 11, 13, 14
24.   Fri 17 Nov SECOND MIDTERM EXAMSECOND MIDTERM EXAM
 Wed 22 Nov no class: Thanksgiving 
    Fri 24 Nov no class: Thanksgiving 
25.Wed 29 Nov6.1eigenvalues and eigenvectors5.1: 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 Dec6.2diagonalizability 
27.Wed  6 Dec7.1; 6.4Jordan form; spectral theorem6.1: 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 Dec7.3matrix 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
Quizzes and participation in class discussion can contribute to your homework score.

Links

University academic linksDepartmental 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


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