TEACHING DOSSIER Contents 1. Introduction 1 1.1. Courses ...

TEACHING DOSSIER

ANILA YADAVALLI

Contents

1. Introduction 11.1. Courses Taught 11.2. Summary of Teaching Philosophy 22. Teaching Evaluations 22.1. Selected Peer and Faculty Observations 22.2. Summary of Student Evaluations 23. Artifacts of Teaching 43.1. Sample Syllabi 43.2. Sample In-Class Assignment 43.3. Sample Study Guide 53.4. Sample Exams 54. Teaching Improvement Activities 55. Conclusion 56. Appendices 5

1. Introduction

In this document, I discuss my career goals and present evidence related to my effectiveness as a collegemathematics instructor. While most of the content highlights positive feedback I have received about myteaching, I acknowledge that I have plenty of room to grow as an instructor. My teaching style is heavilyinfluenced by constructive criticism from faculty observations, regular student feedback, and professionaldevelopment in teaching.

1.1. Courses Taught. Below are the details of all of the courses for which I have served as the instructorof record at North Carolina State University (NCSU). For each of these classes, I developed the syllabus,planned and delivered lectures and in-class activities, wrote exams, quizzes, and homework assignments,and determined final grades. In summer courses and sections with more than 40 students, I managedgraders, TAs and lecture assistants. In sections with fewer than 40 students, I was responsible for gradingexams and homework assignments. Prior to teaching my own courses, I served as recitation leader andlecture assistant for several courses. A full list is included in my CV.

• MA 107: Precalculus I, 3 credit hours.? 186 students, Fall 2018

• MA 114: Introduction to Finite Mathematics, 3 credit hours.? 171 students, Fall 2015

• MA 131: Calculus for Life and Management Sciences, 3 credit hours.? 9 students, Summer I 2015? 118 students, Spring 2017

• MA 141: Calculus I for Scientists and Engineers, 5 credit hours.? 38 students, Spring 2016

• MA 242: Calculus III for Scientists and Engineers, 5 credit hours.? 21 students, Summer II 2016? 51 students, Fall 2016

1

2 ANILA YADAVALLI

? 27 students, Summer I 2017• MA 405: Introduction to Linear Algebra, 3 credit hours.

? 26 students, Spring 2018? 11 students, Summer I 2018

I was also an Apprentice Instructor at a summer program called MathILy during in July 2018, where Itaught linear algebra, combinatorics, and abstract algebra to mathematically advanced high school studentsin a completely inquiry based format.

1.2. Summary of Teaching Philosophy. My goal as an instructor is to be innovative and engaging sothat my students have the opportunity to believe that they can be “math people” regardless of whethertheir math journey has been positive or negative. I create a collaborative and inclusive classroom, andencourage my students to be active learners. I strongly believe in the use of inquiry based learning andwriting activities in the mathematics classroom. I adapt my language based on each course to convey thevalue of I am teaching and the activities we do. I also hold my students accountable for their educationalexperience by promoting effective study habits.

2. Teaching Evaluations

2.1. Selected Peer and Faculty Observations. Section 6, contains the completed observation formsfrom the following faculty and peer observations:

• MA 242: Calculus III for Scientists and Engineers, Fall 2016 : Dr. Ernest Stitzinger, Administratorof Graduate Programs at the time, observed my Calculus III course as part of the Certificate ofAccomplishment in Teaching Program. He was impressed by the passion I brought to the courseand how I connected the content to my research. (See Appendix A, pg. 6).• MA 131: Calculus I for Life and Management Sciences, Spring 2017: Dr. Tye Lidman, Assistant

Professor, was my faculty mentor and observed me twice. He added that he had a high opinion ofmy teaching and was impressed by how I handled such a large lecture hall class. His ratings reflectthat I needed to focus on student engagement, which I agreed with. In my current precalculusclass, which is also a large lecture, I implemented short discovery-based exercises. For example, Ihad students pick their favorite parent function and sketch translations of it, before writing downgeneral rules for translations. (See Appendix B, pg. 11)• MA 405: Introduction to Linear Algebra, Spring 2018 : Dr. Alina Duca, Director of Undergraduate

Programs, was my mentor for the Preparing the Professoriate program and worked with me throughthe semester to help improve my approach to classroom engagement. She pointed out that I wasn’tincluding students in the story, which I addressed by implementing “Discovery Assignments” (see§3.2) in order to create a more active learning environment. (See Appendix C, pg. 12)• MA 405: Introduction to Linear Algebra, Spring 2018 : Dr. Shira Viel, Professor of the Practice at

Duke University, served as my peer observer for the Preparing the Professoriate program. Dr. Vielobserved my class several weeks into the course and was impressed by the collegial atmosphere ofthe classroom and students’ willingness to participate. (See Appendix D, pg. 14)

2.2. Summary of Student Evaluations. At the end of each semester, students at NCSU are asked tofill out teaching evaluations about their instructors. Below is a summary of the statistics corresponding toeach semester I have been an instructor of record.

Su15 Fa15 Sp16 Su16 Fa16 Sp17 Su17 Sp18 Su18Number of students enrolled 9 171 38 21 51 118 27 26 11Response rate 44.4% 73.2% 68.4% 76.2% 64.7% 44.9% 63% 61.5% 90.9%The instructor was enthusias-tic about teaching the course

5.0 3.6 4.8 4.9 4.8 4.4 4.8 4.9 4.7

The instructor consistentlytreated students with respect

5.0 3.5 4.6 4.5 4.8 4.3 4.8 4.9 4.7

Overall, the instructor was aneffective teacher

4.8 2.6 4.0 4.0 4.5 3.8 4.7 4.9 4.8

TEACHING DOSSIER 3

They also have space to leave comments about the strengths and weaknesses of the instructor. Below is asummary of these comments.

“[Anila] was always positive when coming to class and teaching materials. Always took the time to in-dividually help students.” (MA 131, Summer 2015)

“[I would like to nominate my instructor for an Outstanding Teacher Award], because of her strong deter-mination to help everyone understand the material, despite extremely rude and disruptive students.”(MA 114, Fall 2015)

“It’s very obvious that Anila is passionate about teaching math, and you can tell she really enjoysit. This makes it easy to pay attention, and she doesn’t mind helping whenever you approach her.” (MA141, Spring 2016)

“...She knew her material extremely well. She was more than willing to answer questions in class andmade no question seem unimportant. Anytime I needed help outside of class she was beyond willingto give it. I could tell she appreciated those who strive to do better. She was always friendly and happythroughout the course too, even when my classmates were often rude. She was professional but funtoo.” (MA 141, Spring 2016)

“She made it easy to understand difficult topics. She really understood our course load as a studentand would push back assignments when she realized we didn’t have adequate time to complete them. Shewould definitely be my first choice as an instructor if she ever taught any other math class I would betaking.” (MA 242, Fall 2016)

“Anila Yadavalli’s method of explaining the theory or proof behind a concept and showing how it is anal-ogous to something taught in calculus 1 or 2 before showing several examples really helped me gain adeeper appreciation of the field of calculus as a whole.” (MA 242, Fall 2016)

“...Ms. Yadavalli obviously cares a ton about the course and is consistently ebullient and outgoing whenteaching. She’s full of energy, available for outside consultation, and will answer questions about all otherkinds of math, no matter how random. This is the person you want teaching kids: she works with someextra energy that most people don’t seem to have. This does not come at the cost of professionalism; sheis also one of my most focused teachers. From her emails, you would expect her to be much morereserved. I recommended her to all of my friends before learning that she might not be teaching the coursenext year; regardless, whoever ends up having her will luck out....” (MA 242, Fall 2016)

“I believe that the best part of this course was drawing connections between previously 2- dimensionalconcepts or static 3-dimensional concepts and deviating in every possible manner from their well-behavednature to develop a proficiency in manipulating integration and differentiation for real-world applications.Ms. Yadavalli’s enthusiasm about the coursework rubbed off on me, inspiring me to delve intomy readings and to explore several concepts outside of the text, including n-dimensional Taylor Series andtheir applications in integration.” (MA 242, Fall 2016)

“Overall, the instructor did a great job of explaining the material. Coming into the class, I was veryunsure about how well I’d do, and it surprised me how I came out of this calc class with almost all A’s onmy exams. She did an excellent job of providing examples and explaining each objective well. I also likedhow she provided practice problems with solutions before each exam, because it helped me prepare betterand get me used to her exam format. Another strength was that she wrote on the overhead instead of thewhiteboard. I think it made it easier for everyone to see the notes.” (MA 131, Spring 2017)

4 ANILA YADAVALLI

“Ms. Yadavalli was a great teacher this semester. She did well, not only introducing concepts, butbuilding up to them and linking them back to others we had discussed throughout the course.”(MA 405, Spring 2018)

“Absolutely amazing instructor, I would 100% take more classes taught by her. I wish all teachers inthe math department had the passion that she did.” (MA 405, Spring 2018)

“Amazing professor who made sure everyone in class had a full understanding of daily material by provid-ing in class exercises to complete individually or in groups.” (MA 405, Spring 2018)

“Her Moodle page was very organized and helpful. Especially, supplemental resources, dis-cussion forums, and LaTeX sources helped me learn materials effectively.” (MA 405, Summer 2018)

“Great at presenting topics in a way that flowed into one another rather than simply having everythingthrown at us without reason. I really liked how we looked at examples and made observationsourselves before formalizing it into a theorem rather than just listing the theorem immediately, andit helped to work on proofs for some of these theorems in class.” (MA 405, Summer 2018)

I have also received some constructive comments which have I have used to improve how I run mycourse. Many of these comments pertain to the pace of the course and the ease of taking notes. Beloware two constructive comments from an anonymous mid-semester survey I took in my current precalcu-lus course. I follow each comment with a brief description of the actions I have taken to address these issues.

• “Easy examples in class are necessary when learning new content, but then it’s difficult when a[W]eb[A]ssign problem is much more complicated. It might help if we had some easy and a harderexample per each concept in class so that we have more to refer back to.”To address this, I have started to give students examples in which they will purposely run intoan issue. I then use this issue to explain a new topic, and ask them to brainstorm why that issuearose. This way, students still have opportunities for engagement, and we cover a larger breadthof examples without falling behind.

• “I like how many practice problems we do, but it is very challenging to take all of the notes fastenough and understand how the problems are solved without missing something.”To address this, I have started posting my notes online at the end of each week. This way, studentsare still motivated to attend class and write down notes, but they can feel reassured that if theymiss writing something down, they can refer to my notes.

3. Artifacts of Teaching

The following artifacts from my teaching can be found in found in Section 6.

3.1. Sample Syllabi. I have included two sample syllabi that reflect very different teaching experiences.In every syllabus, I include a statement about my commitment to diversity and an inclusive classroom,course policies that keep students accountable for their experience with the course, and tips for success.

? MA 107: Precalculus, Fall 2018 : For this course (current), I am emphasizing good study habitsand becoming comfortable with notation and definitions, which is reflected in the syllabus. (SeeAppendix E, pg. 16)

? MA 405: Introduction to Linear Algebra, Spring 2018 : For this course, I emphasized collaborationand classroom participation. (See Appendix F, pg. 21)

3.2. Sample In-Class Assignment. Appendix G (pg. 26) is an example of an in-class group activity Iused in my linear algebra course to introduce representing linear maps as matrices. Students who referredto the textbook after class commented that completing this activity made the material more digestible.The assignment concluded with students presenting problems from “Exercise 2” at the board.

TEACHING DOSSIER 5

3.3. Sample Study Guide. I am committed to providing students with study guides before each examso they are familiar with the types of questions I write, and so they have a clear set of objectives whenthey begin to study. Appendix H (pg. 30) is a study guide from my linear algebra course.

3.4. Sample Exams.

? MA 141: Calculus I for Scientists and Engineers, Spring 2016 : Students had two hours for thisexam, so I included some questions that were not purely computational (the students were informedof this in advance). (See Appendix I, pg. 35)

? MA 405: Introduction to Linear Algebra, Spring 2018 : Since students were expected to developtheir abstract mathematical thinking, I generally limited computational questions to half of theexam. (See Appendix J, pg. 36)

4. Teaching Improvement Activities

I have participated in the following professional development activities during graduate school at NorthCarolina State University.

• Preparing the Professoriate? Highly selective, university-wide program, consisting of observing a faculty mentor, attending

professional development workshops, completing a teaching improvement project, and teachingan upper division course.

? Faculty mentor: Dr. Alina Duca• Inquiry Based Learning (IBL) Workshop (hosted at Cal Poly, San Luis Obispo)

? Four day workshop in which participants discussed and collaborated IBL activities for a targetcourse.

? Target course: Introduction to Linear Algebra• Certificate of Accomplishment in Teaching (CoAT)

? Program consisted of attending six teaching workshops, two faculty teaching observations, andcompletion of a portfolio project.

? Workshops attended: Introduction to Teaching, Establishing Credibility and Authority in theClassroom, Learning Styles, Motivational Teaching Strategies, Effective Questioning Tech-niques, Classroom Assessment Techniques,

• STEM & Writing Annual Partnership (SWAP)? Selective, semester-long program for STEM graduate students and postdoctoral students con-

sisting of a week-long workshop focused on implementing writing in STEM classes. Participantsthen collaborated with First Year Writing Program faculty, and guest lectured on writing inthe STEM disciplines in a First Year Writing Program class.

5. Conclusion

I am extremely passionate about teaching mathematics, and it is what I want to focus on for the rest ofmy career. I have not been afraid to take on challenging teaching assignments, and I have participated inmany activities with the goal of enhancing my effectiveness as a teacher. I have experience working withlarge class sizes and “math-phobes” as well as advanced undergraduates and mathematics majors. I havehad semesters that haven’t gone as well as I planned, but I have learned from each of those experiencesand tried to improve. I look forward to continuing to develop myself as an instructor.

6. Appendices

Appendix A 6

TEACHING DOSSIER 11

Observer: Tye Lidman; Course: MA 131; Term: Spring 2017

Appendix B

The Graduate School

Preparing the Professoriate Classroom Observation Form

CoATParticipant’sName: AnilaYadavalliCourse observed: MA 405 Dates of observation: January 22, March 16, April 23, 2018 Time of class: 9:35 – 10:25 Observed by: Alina Duca

Needs Improvement Satisfactory Well

Done Not

Observed Introduction: The lesson was introduced in an interesting and effective way, and the goal or purpose of the lesson was clear.

X

Level/Audience: The lesson was presented at an appropriate level for the intended audience. The instructor seemed to know his/her audience.

X

Content: The content was developed in an organized manner, emphasizing important points. Examples or applications helped students relate material to familiar concepts.

X

Purpose of learning activities: The learning activities had an appropriate place in the lesson, and the purpose of each activity was clear.

X

Student participation: The instructor effectively involved the students in the lesson. X

Climate: The instructor established a positive climate of approachability, setting an appropriate tone for the instructor-student relationship.

X

Enthusiasm: The instructor showed enthusiasm for the subject and introduced interesting aspects of the content.

X

Questions: The instructor checked for student understanding and invited student questions. When questions were asked, the instructor responded clearly.

X

Visual aids: The instructor used visuals effectively. Visuals may include the board, handouts, PowerPoint presentations, content projected on a screen, etc.

X

Time management: The pace of the lesson was appropriate, and the instructor managed the time well and brought the lesson to a logical conclusion.

X

For the OBSERVER: Please answer the three discussion questions on the next page.

Appendix C 12

QUESTIONS for the OBSERVER:

1. What were the strengths of this lesson?The presentation style was engaging, new concepts and theoretical results were appropriately motivated before introducing them. The instructor was confident, engaged, and had a good rapport with the students. For each lesson, Anila was very well prepared with hand-outs, good motivating examples, and a lesson structure that followed organically.The mathematical terminology and the explanations were presented at an appropriate level for a 400-level course, with sufficient emphasis on theoretical concepts without making it too overwhelming for the students. She paid great attention to using the mathematical terminology in a proper way, which is not easy to accomplish for a course that is very dense. Anila had very good control of the mathematical content and provided great answers to students questions.

2. What suggestions do you recommend for this instructor?

I would recommend that she experiments with various ways of engaging and interacting the students, work on making better eye contact with them. The questions posed during lectures could have more depth so the students engage a bit more.

Anila used a handout during the first lecture I observed and it felt that the students were not sufficiently engaged in the “story” of the lesson. We discussed this after the observation and the subsequent lectures that I observed had more of a discovery feel to them. I think it would be great for her to work on finding a good balance between providing class materials while presenting the material in a way that the student is eagerly waiting to see what comes next.

3. Are there any additional comments you would like to share with the instructor?

I think Anila did a terrific job with MA 405 this semester, I received positive feedback from students in this class. I have no doubt she will become an amazing educator!

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Appendix D 14

QUESTIONS for the OBSERVER:

1. What were the strengths of this lesson?

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2. What suggestions do you recommend for this instructor? � �

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3. Are there any additional comments you would like to share with the instructor?

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Form adapted with permission from: Pinder, T. and Ross, C. (2007). Strategies for Teaching Assistant and International Teaching Assistant Development. Jossey-Bass.

15

NCSU Department of Mathematics Fall 2018

MA 107-001: Precalculus

Lecture detailsSection 001| MWF 8:30-9:20AM | SAS 2203Instructor: Anila Yadavalli | SAS 3149 | [email protected] | https://go.ncsu.edu/ayadavalliOffice Hours: T: 4-5PM, W: 9:30-10:30AMMoodle page: http://moodle.wolfware.ncsu.edu/TA info:Amanda Reeder | Cox 406 | [email protected] | M, Th: 9:30-10:30AMJack Thompson | TBA| TBA| TBA

Course text & Moodle Precalculus: Algebra and Trigonometry, Burns-Williams, Brenda, 1st Edition. Studentswill purchase the PDF version of the book through WebAssign (http://webassign.ncsu.edu). The cost for thetextbook and access to homework is $72.95. Textbook content will be available on Moodle (https://wolfware.ncsu.edu) once the semester starts. There will also be videos corresponding to each module available that studentscan choose to watch. Students will find the majority of their work and the interactive features of the textbookonline.

Course overview Algebra and basic trigonometry; polynomial, rational, exponential, logarithmic and trigonomet-ric functions and their graphs. Credit for MA 107 does not count toward graduation for students in Engineering,College of Sciences, Bio and Ag Engineering (Science Program), Bio Sci (all options), Math Edu, Sci Edu, Textiles,and B.S. degrees in CHASS. Credit is not allowed for both MA 107 and MA 111

Grading Policy Your final grade will be determined as follows:

Module Pre-tests & Try-Its 8 %Four Exams 52 % (13% each)Homework 10 %Final Exam 30 %

The grading will be assigned on a 10-point scale.

A+ ≥ 99 A 98.9 - 93 A- 92.9 - 90B+ 89.9 - 87 B 86.9 - 83 B- 82.9 - 80C+ 79.99 - 77 C 76.9 - 73 C- 72.9 - 70D+ 69.9 - 67 D 66.9 - 63 D- 62.9 - 60

F < 60

To maintain fairness and consistency, grades will not be rounded or “bumped”, even in borderline cases.

Module Pre-tests & Try-its Module Pre-tests are designed to test your familiarity with the material before itis taught and give you an idea of what will be covered in that module–it is okay if you cannot answer all of thequestions! The Pre-tests are due at 8:20AM the day we are scheduled to begin a new module (see ClassSchedule for these dates). The Try-Its are designed to test your understanding of the material after we have learnedit and give you a chance to earn back points on your Pre-tests. All Pre-tests and Try-Its can be found on theMoodle page. Any points missed on a Pre-test can be earned back by completing additional Try-Its.

Homework Assignments Homework will be assigned, submitted, and graded online via the WebAssign homeworkportal. You must register and pay for the use of WebAssign at http://webassign.ncsu.edu. Students areencouraged to form groups and work on the homework together as long as everyone gains an equal understandingof the material.

Homework due dates: September 5, September 18, October 18, November 8, November 29. Homeworkmust be submitted by the 11:59PM deadline on the due date to avoid penalty.

Four Exams There will be four in-class, closed book, exams on the following dates. (Module numbers maychange.)Test 1 (Modules 1-6): Wednesday, September 19Test 2 (Modules 7-11): Friday, October 19

Appendix E16


Test 3 (Modules 12,13,15): Friday, November 9Test 4 (Modules 14,16-18): Friday, November 30

Please do not schedule any personal travel, doctor’s appointments, interviews etc. on any of these dates. A scientificcalculator (no graphing capability) will be permitted on each exam.

Final Exam The final exam is mandatory, cumulative and will be held in the usual classroom on December 12,2018 from 8:00-11:00AM.

Test Make-Up Policy If you miss an exam for undocumented reasons, you will receive a zero. All anticipatedabsences must be excused prior to the test date. These include university duties or trips (certified by an appropriatefaculty or staff member), required court attendance (certified by the Clerk of Court), military duty (certified bythe student’s commanding officer), or religious observances (certified by the Department of Parent and FamilyServices 515-2441). Emergency absences must be reported as soon as possible once returning to class and mustbe appropriately documented (illness by an attending physician or family emergencies by Parent and FamilyServices). Students who miss a test and have a university-approved excuse will only be given a make-up exam ifthey submit appropriate documentation. All make-up exams will take place on Friday December 7, 2018 from8:30-9:20AM, with no exceptions.

Corrections to the grading If you believe an error has been made in grading, please write a statement makingyour case and bring it to your TAs within 24 hours. Do not alter the original work!

Bluebooks All students must submit 6 small bluebooks to either TA by Friday, September 14, 2018. Studentswho do so will automatically get 2 bonus points on the first exam. Students who do not will lose 2 points on thefirst exam, and each subsequent exam until the bluebooks are submitted.

Attendance in compliance with university policy, attendance will be recorded daily. Attendance will be takenusing a seating chart; if you sit in the wrong seat you will be marked absent, no exceptions. If you need to changeyour seat at any time during the semester, you must notify both TAs. Attendance does not factor in to your coursegrade. Any student with 5 or fewer absences will have their lowest exam score replaced by their final exam grade,if it is to their advantage. If you miss class, it is your responsibility to obtain notes and catch up. Office hours willnot be used to reteach material.

Students with disabilities Reasonable accommodations will be made for students with verifiable disabilities. Inorder to take advantage of available accommodations, students must register with Disability Services for Studentsat 1900 Student Health Center, Campus Box 7509, 515-7653. For more information on NC State’s policy on workingwith students with disabilities, please see the Academic Accommodations for Students with Disabilities Regulation(REG02.20.1) Students registered with the DSO must schedule all five exams during the first weekof class! I am unable to accommodate any students in my office.

Academic Integrity Statement and Academic Dishonesty Academic dishonesty is the giving, taking, orpresenting of information or material by a student that unethically or fraudulently aids oneself or another on anywork which is to be considered in the determination of a grade or the completion of academic requirements or theenhancement of that student’s record or academic career.

All students are expected to comply with the University policy on academic integrity. Any suspected violationswill be reported.

Non-Discrimination Policy

“NC State University promotes equal opportunity and prohibits discrimination and harassment basedupon one’s age, color, disability, gender identity, genetic information, national origin, race, religion, sex(including pregnancy), sexual orientation and veteran status.” (NCSU Non-Discrimination Policy)

In my classroom, diversity and individual differences are respected, appreciated, and recognized as a source ofstrength. Students in this class are encouraged and expected to speak up and participate during class, and tocarefully and respectfully listen to each other. Every member of this class must show respect for every othermember of this class, so that everyone feels comfortable participating. Any attitude or belief that implies thatone person or group of people is superior to another is not welcome. Such beliefs are destructive to our classroomcommunity and will hinder our ability to learn from each other.

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Additional Resources

• Free tutoring is available through the department Multi Media Center (SAS 2105) on a drop-in basis, Mon-day through Friday from 8:00am to 5:00pm. For more information, see https://www.math.ncsu.edu/mmc/

tutoring.php

• One-on-one free tutoring is available through the Undergraduate Tutorial Services. For more information,see http://www.ncsu.edu/tutorial_center

• For supplemental reading, I suggest using the free online book Openstax, provided by Rice University. Thereis a link to the book on the Moodle Page.

Instructor’s commitment You can expect your instructor to be courteous, punctual, well organized, and pre-pared for lecture and other class activities; to answer questions clearly and in a non-negative fashion; to be availableduring office hours or to notify you beforehand if they are unable to keep them; to provide a suitable guest lecturerwhen they are traveling; and to grade uniformly and consistently according to the posted guidelines.

Statement of Encouragement Learning math is like learning a new language, and is best done collaboratively.Please work with your classmates (while keeping academic integrity in mind), and do not get discouraged if you donot understand a topic immediately. There is no such thing as a “math brain,” nor is mathematical ability innate,so each student should be able to succeed by putting in the necessary effort. Please be aware that everyone in thisclass has different backgrounds and will learn and progress at different paces; you should avoid comparing yourselfto others. Expect to come to class each day, be engaged, and be prepared to discuss problems with your peers.Finding a support system will improve your experience in the course greatly.

How to Win at MA107? Before Class: Watch videos, read text and take Pre-test for the upcoming Module.? During Class: Be present, take notes, and actively participate by asking questions.? After Class: Keep up with the WebAssign homework, do the Try-Its for additional practice and to earn backlost Pre-test points.? When you feel yourself falling behind: come talk with me during office hours, drop by the MMC, or set upan appointment at the Tutorial Center.

Course Policies

• You do not need an appointment to attend my scheduled office hours. If you cannot attend the scheduledoffice hours, email me to set up an appointment as opposed to just “showing up” at my office.

• The best way to contact me is by email. Please make sure that you include your name and the course numberin the subject of the e-mail. Do not respond directly to emails that I send to the entire class, as they will getarchived and I may not see them. If you have general questions about the course, please check the syllabusand Moodle page before sending me an email. Only thoughtful (addressed and signed) and detailed emailswill receive a response. I cannot always guarantee a response to emails, so it is in your best interest to seekout the information on your own before sending me an email.

• Your email address registered with the NCSU online directory will be used for announcements associatedwith this class. It is your responsibility to maintain a valid email address and check your inbox regularly.

• The test grades will be recorded on Moodle. Please notify a TA immediately if you notice any discrepanciesin your grades. Keep all your quizzes and tests for future reference.

• Please check Moodle regularly, as it will be continuously updated with announcements, any changes in theschedule, homework problems, solutions, review sheets, and other additional course materials.

• Please mark the test dates on your calendar and do not schedule medical appointments, interviews, personaltravel, etc. on the test dates.

• The use of cell phones, laptops, and other electronic devices is not permitted in class unless approved by theinstructor. Students who fail to comply will be marked absent and may be asked to leave class.

• If you plan to come to office hours, be prepared with substantial questions. Office hours will not be used tosimply reteach material if you have fallen behind.

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Tentative Course Schedule Any class day with a starred Module (i.e. *Mod 1) indicates that the Pre-Test quizfor that Module is due at 8:20am on that day unless otherwise stated in class.

Date Day Text Topic

Aug 22 Wed Intro to CourseAug 24 Fri *Mod 1 Functions overview, domain, range, difference quotientAug 27 Mon *Mod 2 Algebra of functionsAug 29 Wed *Mod 3 Graphs of functionsAug 31 Fri Mod 1-3 Review of Modules 1-3Sep 3 Mon Labor Day - No ClassSep 5 Wed *Mod 4 Piece-wise defined functionsSep 7 Fri *Mod 5 Graphing by translationsSep 10 Mon Mod 4-5 Finish material from Modules 4-5Sep 12 Wed *Mod 6 Writing functions for modelingSep 14 Fri Module 6 continuedSep 17 Mon Mod 1-6 Test Review: Modules 1-6Sep 19 Wed Test 1: Modules 1-6Sep 21 Fri *Mod 7 Quadratic functionsSep 24 Mon Continue Module 7Sep 26 Wed *Mod 8 PolynomialsSep 28 Fri Finish material from Modules 7-8Oct 1 Mon *Mod 9 Rational functions; domain and vertical asymptotesOct 3 Wed Continue with Mod 9; end behavior and more asymptotesOct 5 Fri Fall Break - No ClassOct 8 Mon *Mod 10 Composing functionsOct 10 Wed Finish material from Module 10Oct 12 Fri *Mod 11 Inverse functionsOct 15 Mon Finish material from Module 11Oct 17 Wed Mod 7-11 Test Review: Modules 7-11Oct 19 Fri Test 2: Modules 7-11Oct 22 Mon *Mod 12 Exponential functionsOct 24 Wed Finish material from Module 12Oct 26 Fri *Mod 13 LogarithmsOct 29 Mon Continue Module 13Oct 31 Wed Mod 17-18 Review material from Mod 12-13Nov 2 Fri *Mod 15 Logarithmic functionsNov 5 Mon Finish material from Module 15Nov 7 Wed Mod 12, 13, 15 Test Review: Modules 12, 13, 15Nov 9 Fri Test 3: Modules 12, 13, 15Nov 12 Mon *Mod 14 & *Mod 16 Using exponential and logarithmic modelsNov 14 Wed *Mod 17 Angles and arcsNov 16 Fri Finish material from Module 17Nov 19 Mon *Mod 18 Right triangle trigonometry

Nov 21-23 Wed-Fri Thanksgiving Break - No ClassNov 26 Mon Finish material from Module 18Nov 28 Wed Mod 14, 16-18 Test Review: Modules 14, 16-18Nov 30 Fri Test 4: Modules 14, 16-18Dec 3 Mon Final Exam ReviewDec 5 Wed Final Exam ReviewDec 7 Fri Final Exam Review

Dec 12 Fri Final Exam: All Modules

19


General Tips for Studying Math

Here are a few tips that I have found useful when studying for a math course.

• Notation: Understand all notation and be able to translate from a sentence into mathematical notation andvice versa. When encountering new notation, ask yourself the following questions:

? What type of mathematical object is this notation describing? (Ex. A function? A set? An operation?etc.)

? Have I learned other notation to describe this object? (Ex. a−1 = 1a ).

? What parts of this notation are static, and what parts can I change? (Ex. f(x) and g(x) both denotefunctions).

• Definitions: Know your definitions! Be able to use them in a sentence or example. Be able to use them whenthey are used in other definitions. Be able to use them to answer questions or solve problems. When learninga new definition,

? Highlight or underline all of the defining qualities.

? Make sure you know all of the terms in the definition.

? Come up with your own examples and nonexamples.

? Example: Definition. A panda bear is a bear that eats bamboo and is black and white.

∗ The defining qualities of a panda bear are that it is a bear, it eats bamboo, and is black and white.

∗ To understand what a panda bear is, we need to know what a bear is, what bamboo is, and whatthe colors black and white are.

∗ Example of a panda bear:Nonexample of a panda bear:

• Come to class: Be engaged, ask questions and actively think about any in class activities or problems.Learning material outside of class takes twice as long.

• Practice! Practice! Practice!

– Go back through the examples that were done in class. If possible, try them on your own and then checkyour solution against the notes. Make sure each of your steps makes logical sense even if you got thecorrect answer! If you have any questions, talk to a classmate, seek help from the instructor/TAs orother resources.

– Do not fall behind. The sooner you complete your homework after learning a topic, the more preparedyou will be to learn new material.

– Do as many problems as you can, and be patient. Don’t get frustrated if you don’t understand somethingright away.

– Work with a peer or form a study group. In mathematics, it is very helpful to bounce ideas off of eachother.

– Not all examples are the same: if a problem looks familiar, don’t try to memorize or replicate a solutionyou have already seen. Understand what the problem is asking and keep note of how it is similar and/ordifferent from other problems you have seen.

• Other tips: Put your phone on silent and out of sight! If you listen to music, try music without words.Mathematics requires deep concentration and frequent interruptions will hinder the process of completing asolution; Come to office hours and get to know the instructor/TAs; Try rewriting your lecture notes afterclass; Find a friendly face and form study groups early on.

20

NCSU Department of Mathematics Spring 2018

MA 405. Introduction to Linear Algebra and Matrices

Lecture detailsSection 001| MWF 9:35 - 10:25| SAS 2225Instructor: Anila Yadavalli | SAS 3213 | [email protected] | https://anilayadavalli.wordpress.ncsu.edu/Office Hours: M. 10:30-11:30AM, Th. 10:00-11:00AM, and by appointmentMoodle page: http://moodle.wolfware.ncsu.edu/

Course textLinear Algebra Done Right, by Sheldon Axler, Springer International Publishing : Imprint: Springer, 2015ISBN: 9783319110806 - available through NCSU libraries.

Helpful (but not required) textsLinear Algebra and its Applications, by David C. Lay, Pearson Education International, any editionISBN: 0321149920Introduction to Linear Algbera with Applications, by Jim DeFranza and Daniel Gagliardi, Waveland Press, Inc.ISBN: 1478627077

Course overviewPrerequisite: MA 241 (Co-requisite MA 242)

Linear Algebra provides one of the cornerstones for much of modern Mathematics, and has important applica-tions in Physics, Engineering, and Economics. The main purpose of this course is to introduce the basic conceptsfrom linear algebra, explain the underlying theory, the computational techniques, and study how these conceptsand results can be productively used in other areas of mathematics and physical sciences, especially in appliedmathematics where multivariable models are involved. Among the topics covered in this course will be: solvingsystems of linear equations using Gauss elimination, row echelon form, determinants, vector spaces, linear indepen-dence, bases, dimension, linear transformations, orthogonality, eigenvalues, and reduction of matrices to diagonalforms.

The subject involves a mixture of both the practical and the theoretical, and will provide in particular a goodintroduction to mathematical proofs. For this reason, the course is considered to be a difficult one in undergradu-ate mathematics, and the student should be prepared to invest considerable amount of time in understanding theclass material and doing homework.

Credit is not allowed for both MA 305 and MA 405.

Learning ObjectivesUpon successful completion of this course, students will be able to:

1. Use Mathematical Notation and Terminology.

2. Understand and Communicate the Fundamental Concepts of Linear Algebra.

3. Identify and Utilize Linear Algebra Tools.

4. Develop Cognitive Skills.

Grading PolicyThe grading will be assigned on a 10-point scale:

A+ ≥ 99 A 98.9 - 93 A- 92.9 - 90B+ 89.9 - 87 B 86.9 - 83 B- 82.9 - 80C+ 79.99 - 77 C 76.9 - 73 C- 72.9 - 70D+ 69.9 - 67 D 66.9 - 63 D- 62.9 - 60

F < 60

21Appendix F


Your final grade in this course will be determined as follows:

Homework = 30 %Three midterms = 42 %Final Exam = 21 %Classroom Participation = 7 %

Three Exams 42%There will be three closed book, in-class tests on February 2, March 2, and April 13.

Final Exam 21%The final exam is mandatory, cumulative and will be held in the usual classroom on Monday, April 30, 2017,8:00-11:00AM.

Homework Assignments 30%Homework will be assigned and collected approximately every two weeks. The homework assignments will beavailable on Moodle. Homework should be written up neatly or typed in LaTeX. All solutions must be completelyjustified. Students are encouraged to collaborate on homework as long as everyone in the group gains a thoroughunderstanding of the solution. Each student must write up the solution in their own words based on their ownunderstanding, keeping academic integrity in mind. Please list the names of everyone you have collaborated withat the top of your assignment when you turn it in. Additional problems will be available through WeBWorK, butwill not count for a grade. It is in your best interest to complete these optional problems as they are fair game forexams.

Participation/Presentations 7%This course will rely heavily on classroom collaboration and participation. Students will have many opportunitiesto present problems at the board or in small groups. Everyone must present at least one problem during thesemester. Students will be expected to engage actively, respectfully, and inclusively with their classmates duringand outside of class. Group activities are not a time to relax, “chit-chat,” or let someone else do all the work.

Corrections to the gradingIf you believe an error has been made in grading, please write a statement making your case and bring it to yourinstructor within 24 hours. Do not alter the original work!

Test Make-Up PolicyIf you miss an exam for undocumented reasons, you will receive a zero. All anticipated absences must be excusedprior to the test date. These include university duties or trips (certified by an appropriate faculty or staff member),required court attendance (certified by the Clerk of Court), military duty (certified by the student’s commandingofficer), or religious observances (certified by the Department of Parent and Family Services 515-2441). Emergencyabsences must be reported as soon as possible once returning to class and must be appropriately documented(illness by an attending physician or family emergencies by Parent and Family Services). If you are sick on a testday and decide not to come to class, go to the health center or other medical facility. Students who miss a testand have a university-approved excuse must submit appropriate documentation.

Attendance Poor attendance serves as its own penalty because material takes much longer to learn independently.You are responsible for keeping up with missed work so that you do not fall behind. Office hours will not be utilizedto re-teach material presented in class. Poor attendance may negatively impact your participation grade.

Students with disabilities“Reasonable accommodations will be made for students with verifiable disabilities. In order to take advantageof available accommodations, students must register with Disability Services for Students at 1900 Student HealthCenter, Campus Box 7509, 515-7653. For more information on NC State’s policy on working with students withdisabilities, please see the Academic Accommodations for Students with Disabilities Regulation (REG02.20.1)”Please schedule all DSO exams at least two days prior to the first exam.

22


Academic Integrity Statement and Academic Dishonesty

“Academic dishonesty is the giving, taking, or presenting of information or material by a studentthat unethically or fraudulently aids oneself or another on any work which is to be considered inthe determination of a grade or the completion of academic requirements or the enhancement of thatstudent’s record or academic career.” (NCSU Code of Student Conduct)

All students are expected to comply with the University policy on academic integrity. Any suspected violationswill be reported.

Adverse WeatherAnnouncements regarding scheduled delays or the closing of the University due to adverse weather conditions willbe broadcast on local radio and television stations and posted on the University homepage.

Non-Discrimination Policy

“NC State University promotes equal opportunity and prohibits discrimination and harassment basedupon one’s age, color, disability, gender identity, genetic information, national origin, race, religion, sex(including pregnancy), sexual orientation and veteran status.” (NCSU Non-Discrimination Policy)

In my classroom, diversity and individual differences are respected, appreciated, and recognized as a source ofstrength. Students in this class are encouraged and expected to speak up and participate during class, and tocarefully and respectfully listen to each other. Every member of this class must show respect for every othermember of this class, so that everyone feels comfortable participating. Any attitude or belief that implies thatone person or group of people is superior to another is not welcome. Such beliefs are destructive to our classroomcommunity and will hinder our ability to learn from each other.

Instructor’s commitmentYou can expect your instructor to be courteous, punctual, well organized, and prepared for lecture and other classactivities; to answer questions clearly and in a non-negative fashion; to be available during office hours or to notifyyou beforehand if they are unable to keep them; to provide a suitable guest lecturer when they are traveling; andto grade uniformly and consistently according to the posted guidelines.

Statement of EncouragementLearning math is like learning a new language, and is best done collaboratively. Please work with your classmates(while keeping academic integrity in mind), and do not get discouraged if you do not understand a topic immediately.There is no such thing as a “math brain,” nor is mathematical ability innate, so each student should be able tosucceed by putting in the necessary effort. Please be aware that everyone in this class will learn and progress atdifferent paces, and you should avoid comparing yourself to others.

23


Course Policies

• You do not need an appointment to attend my scheduled office hours. If you cannot attend the scheduledoffice hours, email me to set up an appointment as opposed to just “showing up” at my office.

• You do not need to e-mail me if you are missing class.

• The best way to contact me is by email. Please make sure that you include your name and the course numberin the subject of the e-mail. Do not respond directly to emails that I send to the entire class, as they will getarchived and I may not see them.

• Your email address registered with the NCSU online directory will be used for announcements associatedwith this class. It is your responsibility to maintain a valid email address and check your inbox regularly.

• The test grades will be recorded on Moodle. Please notify me immediately if you notice any discrepancies inyour grades. Keep all your quizzes and tests for future reference.

• Please check Moodle regularly, as it will be continuously updated with announcements, any changes in theschedule, homework problems, solutions, review sheets, and other additional course materials.

• Please mark the test dates on your calendar and do not schedule medical appointments, interviews, personaltravel, etc. on the test dates.

• Be respectful to your peers and to your instructor. The use of cell phones and laptops is not permitted inclass unless approved by the instructor. Students who fail to comply may be asked to leave class.

• If you are registered to take exams at the DSO, please schedule all four exams at least two days beforeExam 1.

24


MA405 Tentative Schedule

• Week 1. Euclidean Vector Spaces. Vector Spaces. Examples. (1.A, 1.B)

• Week 2. Subspaces. Span (1.C, 2.A)

• Week 3. Linear Independence, (2.A)

• Week 4. Basis & Dimension (2.C)

• Week 5. Row space/Column space/Null Space/ Rank Theorem. (Instructor Notes)

• Week 6. Coordinates. Change of Basis. (Instructor Notes)

• Week 7. Linear Maps. Kernel, Range or a Linear Map (3.A, 3.B)

• Week 8. Linear Maps continued (3.B)

• Week 9. Matrix Theory. Invertible matrices. (3.C)

• Week 10. Isomorphisms. (3.D)

• Week 11. Eigenvalues & Eigenvectors (Instructor Notes)

• Week 12. Eigenspaces. Diagonalization. Similarity (Instructor Notes)

• Week 13. Inner Product Spaces. Orthonormal Bases. Gramm-Schmidt process. (6.A, 6.B)

• Week 14. Orthogonal Complements. Least squares approximation. (Instructor Notes)

• Week 15. Diagonalization of symmetric matrices. Singular Value Decomposition. (Instructor Notes)

25

Discovery Assigment #6 MA 405-001

1 Review

• If A ∈Mm×n, then the transformation T : Rn → Rm defined by v 7→ Av is a linear transformation.

• If T : Rn → Rm is a linear transformation, then there exists a matrix A ∈Mm×n such that T (v) = Av.

• If V is a vector space, and dim(V ) = n, V ∼= Rn (in particular, the map v 7→ [v]B for some ordered

basis B is an isomorphism.)

• If V and W are two vector spaces and dim(V ) = dim(W ) = n, then V ∼= W .

2 Exercise 1: Review

1. Let B be the standard basis for R3. Suppose T : R3 → R2 is defined by T (e1) =

1

1

, T (e2) =

−1

2

and T (e3) =

0

1

. Find a description for T (v) in the form T (v) = Av.

2. Use part (1) to find T (v) if v =

1

2

−1

3. Let B′ = {u1,u2,u3} =

1

0

0

,

1

1

0

,

1

1

1

be another basis for R3. Find T (u1), T (u2), and T (u3).

4. Let [u]B′ =

1

2

3

. Find T (u) (Hint: your result from #3).

3 Exercise 2: Representing any Linear Transformation as a Ma-

trix

Let V and W be vector spaces with dim(V ) = 3 and dim(W ) = 2. Let B = {v1,v2,v3} be an ordered basis

for V and B′ = {w1,w2} an ordered bases for W . Let v ∈ V such that [v]b =

c1

c2

c3

.

26Appendix G

1. Write v as a linear combination of vectors in B.

2. Find T (v) in terms of T (v1), T (v2), T (v3).

3. Since T (v1), T (v2), T (v3) ∈W , we can find a coordinate vector for each T (vi) with respect to B′. Let

[T (vi)]B′ =

a1ia2i

, i = 1, 2, 3.

Write each T (vi) as a linear combination of the vectors in B′.

4. Write T (v) as a linear combination of the vectors in B′.

5. Find [T (v)]B′ .

6. Write a matrix A such that A[v]B = [T (v)]B′ .

4 Definitions, Theorems and Examples

Definition 4.1. Let V and W be vector spaces

Theorem 4.2. Let V and W be finite dimensional vector spaces

Example 4.3. If V is a vector space with ordered bases B and B′, and I : V → V is the identity map,

Then [I]B′

B is just the transition matrix [I]B′

B .

2

27

Example 4.4. Define the linear operator T : R2 → R3 by

T (v) = T

x1x2

=

x2

x1 + x2

x1 − x2

and let

B =

1

2

,

3

1

, B′ =

1

0

0

,

1

1

0

,

1

1

1

be ordered bases for R2 and R3 respectively.

1. Find the matrix [T ]B′

B .

2. Let v =

−3

−2

. Find T (v) using the matrix from part (a). Verify your answer by finding T (v) directly.

Example 4.5. Define a linear transformation T : P2 → P3 by

T (f(x)) = x2f ′′(x)− 2f ′(x) + xf(x)

Find the matrix representation of T relative to the standard bases for P2 and P3. Use it to find T (x2−3x+1).

3

28

Example 4.6. Let V and W be finite dimensional vector spaces with ordered bases B and B′ repsectively.

Let S and T be linear transformations from V to W . Find the matrices for S + T and αT relative to B and

B′.

Example 4.7. Let U , V and W be finite dimensional vector spaces with ordered bases B, B′ and B′′

repsectively. Let T : U → V and S : V → W be linear transformations. Find the matrix for ST relative to

B, B′ and B′′.

5 Exercise 2

1. Let D : P3 → P3 be the linear operator defined by D(p(x)) = p′(x).

(a) Find the matrix of D relative to the standard basis B = {1, x, x2, x3}. Use it to find

D(1− x+ 2x3).

(b) Can we use this matrix to find p′′(x)? If so, how?

2. Let S2×2 =

a b

b c

: a, b, c ∈ R

and T : P2 → S2×2 be defined by ax2 + bx+ c 7→

a b

b c

.

(a) Show that T is an isomorphism.

(b) Find [T ]B′

B and [T−1]BB′ if B = {1, x, x2} and B′ =

0 0

0 1

,

0 1

1 0

,

1 0

0 0

. (Remember

that if T : V →W , then T−1 : W → V .)

3. Let T : M2×2 →M2×2 be defined by A 7→ At. Let B = {E11, E12, E21, E22}. Find [T ]BB .

4

29

Study Guide for Exam 3

MA 405 - 003

1 Students should be able to....

• Compute the null space and column space of a matrix.

• Determine the rank and nullity of a matrix.

• Apply the 12-part theorem to draw conclusions about n× n matrices.

• Find the coordinates of vector with respect to a given ordered bases.

• Find and use the transition matrix from one ordered basis to another.

• Give examples of linear maps.

• Determine if a function between vector spaces is a linear map.

• Prove general statements about linear maps.

• Find a matrix corresponding to linear maps between Euclidean n-spaces.

• Find a basis for the null space and range of a linear map.

• Determine if a linear map is injective and/or surjective (both explicitly and using dimension).

• Determine if two vector spaces are isomorphic, and if so construct an isomorphism between them.

• Find and use the matrix of a linear transformation.

• Evaluate the image of a linear transformation using its matrix.

• Determine if a number is an eigenvalue for a matrix.

• Find eigenvalues of an n× n matrix and the corresponding eigenspaces.

• Find the characteristic polynomial and equation of a matrix.

(Scroll down for problems)

30Appendix H

of 5

2 Suggested Problems

1. Find null(A) and col(A). State the rank and nullity of A as well.

(a) A =

1 1 1

2 4 −1

0 2 −3

(b) A =

4 0 −1 0

0 1 −1 0

2 3 0 0

(c) A =

2 −1 0

1 0 1

1 1 0

2. Let S = {e1, e2} be the standard ordered basis for R2 and let B =

1

0

,

−

12

12

(a) Find [I]BS .

(b) Find the coordinates of a vector v =

ab

∈ R2 with respect to B.

(c) Draw the rectangle in R2 with vertices at the points (1, 2), (1, 4), (4, 2), (4, 4).

(d) Find the coordinates of each of the vertices with respect to the basis B.

3. Let B1 = {u1,u2,u3} and B2 = {v1,v2,v3} be two ordered bases for a vector space V such that

u1 = −v1 + 2v2,u2 = −v1 + 2v2 − v3 and u3 = −v2 + v3.

(a) Find the transition matrix [I]B2

B1.

(b) Find [u]B2 if u = 2u1 − 3u2 + u3.

4. Let V be a vector space and B and B′ two ordered bases for V . Let [I]B′

B =

1 −1 0

0 1 −1

0 0 1

. Find

[I]BB′ .

5. Determine if the following functions T are linear transformations. If so, determine if they are injective,

surjective, both or neither.

(a) T : P3 → R2, ax3 + bx2 + cx+ d 7→

−a− b+ 1

c+ d

.

Cont.

31

of 5

(b) T : R3 → R2,

x

y

z

7→

x

2 − 1

y + z

.

(c) T : P3 → P3, p(x) 7→ x+ p(x).

(d) T : M2×2,→ R A 7→ det(A).

(e) T : R2 → R2, T

xy

=

1 0

−2 −3

xy

.

6. Suppose that T : V → V is a linear operator and {v1,v2, . . . ,vn} is linearly dependent. Show that

{T (v1), T (v2), . . . , T (vn)} is a linearly dependent set.

7. Give an example of a linear map from P2 to R3 that is not an isomorphism.

8. Let T1 : V → R and T2 : V → R be linear transformations. Show that T : V → R2 defined by

T (v) =

T1(v)

T2(v)

(1)

is a linear transformation.

9. Find a basis for the range of the following linear transformations. Then state the dimension of the null

space of T . Conclude whether T is injective or not.

(a) T : R3 → R3 defined by

x

y

z

7→

x

y

0

.

(b) T : P3 → P3 defined by p(x) 7→ p′′(x) + p′(x) + p(0).

10. Give an example of a linear transformation T : R3 → R2 such that R(T ) = R2.

11. Show that if T : V → V is a linear operator with R(T ) = N(T ), then dim(V ) is even.

12. Let T : M2×2 →M2×2 be defined by A 7→ A+At. Find a basis for N(T ).

13. Find an isomorphism from R4 to P3. Justify that it is an isomorphism

14. Find an isomorphism from M2×2 to P3. Justify that it is an isomorphism.

15. Let T : Rn → Rn be defined by T (v) = A(v) for some A ∈ Mn×n. Show that T is an isomorphism if

and only if A is invertible. (Hint: The 12 part theorem will help).

16. Let A =

2 −1

1 0

. Let T : M2×2 → M2×2 be defined by T (M) = AM − MA. Find [T ]BB if

B = {E11, E12, E21, E22}. Use it to evaluate T

3 10

−2 7

.

Cont.

32

of 5

17. Find all eigenvalues of A and their corresponding eigenspaces.

(a) A =

1 −2

0 1

(b) A =

2 1 2

0 2 −1

0 1 0

18. Let A be an invertible matrix with eigenvalue λ. Find an eigenvalue for A−1.

19. Let A be an n× n matrix. Show that if A is not invertible if and only if λ = 0 is an eigenvalue for A.

20. Let λ be an eigenvalue for A. Show that if λ is an eigenvalue for A then λn is an eigenvalue for An.

What can you conclude about the eigenvectors?

21. Let C = B−1AB. Let v be an eigenvector for C with eigenvalue λ. Find an eigenvalue and eigenvector

for A.

22. Give an example of a linear map from R3 to M2×2 that has a trivial null space.

3 True or False (Prove or give a counterexample)

1. Let A be an invertible n× n matrix. A has a trivial null space.

2. Let A be an n× n matrix with nonzero determinant. The columns of A form a basis for Rn.

3. Let A be an 2× 4 matrix. Then rank(A) ≤ 4.

4. Let A be an n× n matrix that is not invertible. Then col(A) = Rn.

5. Let A be an n× n matrix with det(A) = 2. The linear map T : Rn → Rn defined by T (v) = Av is an

isomorphism.

6. Let A be a 4× 3 matrix. Let T : R3 → R4 be defined by T (v) = Av. T is surjective.

7. Let T : P2 →M2×2 be a linear map. Then range(T ) forms a basis for M2×2.

8. It is possible to find an isomorphism from P2 to M2×2.

9. It is possible to find a surjective linear map from P2 to M2×2

10. It is possible to find an injective linear map from P2 to M2×2.

11. Let T : V → W be an injective linear transformation and B = {v1,v2, . . .vn} ⊆ V a linearly

independent set. {T (v1), T (v2), . . . , T (vn)} ⊆W is a linearly independent set.

Cont.

33

of 5

12. The linear map D : P4 → P3 defined by D(p(x)) = p′(x) injective.

13. Let T : M2×2 →M2×2 be defined by A 7→ A2 −A. T is a linear map.

14. Let TV → W be a linear transformation and B = {v1,v2, . . . ,vn} a basis for V . If T is surjective

then {T (v1), T (v2), . . . , T (vn)} forms a basis for W .

15.

0

1

is an eigenvector for

−4 0

3 −5

.

The End.

34

MA 141 Exam 3 Version A 17 March 2016

Write your full name and “YADAVALLI” on the front of the bluebook provided. Pleasewrite responses clearly in your bluebook. Show all of your work. No credit will begiven for unjustified solutions. You may use a scientific calculator only. You have1 hour and 50 minutes. Please fold your exam and put it in your bluebook whencomplete. Please turn off and put away all cell phones and electronics.

Name:

1. (10 PTS) A 16-ft ladder slides down a wall. Let h(t) denote the height of the ladder at time t and letx(t) denote the distance from the wall to the bottom of the ladder at time t. Suppose the top is slidingdown the wall at a rate of -4 ft/s. How fast is the bottom of the ladder moving when h(t) = 12. (Youdo not have to simplify your answer)

2. (10 PTS) The radius of a sphere is increasing at a rate of 4mm/s. How fast is the volume increasingwhen the diameter is 80 mm? (You do not have to simplify your answer)

3. (5 PTS) Use linearization to approximate ln(1.07). You must show your work. No credit will begiven for using only a result from your calculator.

4. (10 PTS) Find the linearization of f(x) =√

1 + x at a = 8. Use it to approximate√

9.01.

5. (10 PTS) Find the critical points of f(x) = (2x−x2)ex. Classify them using the Second Derivativetest. Hint: the domain of ex is (0,∞).

6. (15 PTS) Sketch the graph of f(x) = 3x4−8x3+6x2+1 by finding and classifying all maxima, minima,and inflection points.

7. (15 PTS) A box with square base is constructed out of two different types of metal. The metal forthe top and bottom costs $1/ft2, and the metal for the sides costs $2/ft2. Find the dimensions thatminimize cost if the box has a volume of 20 ft3.

8. (10 PTS) Evaluate the following limits:

(a)

limx→∞

x2

ex

(b)

limx→∞

x1x

9. (5 PTS) Use Newton’s Method with x1 = 2.5 to find the fourth approximation of the root of f(x) =x2 − 7.

10. (10 PTS) Sketch a possible graph of a function with the following properties:

(a) f ′(x) = 0 when x = −1, 0, 3.

(b) f ′(x) > 0 when 0 < x < 3 and f ′(x) < 0 when x > 3.

(c) f ′′(−1) = 10

(d) f ′′(x) = 0, f ′′(x) < 0 when x > 0, and f ′′(x) > 0 when x < 0.

11. (3 PTS) Bonus: Evaluate the following limit, assuming f is differentiable and f ′ is continuous:

limx→a

f(x)− f(a)

x− a.

35Appendix I

MA 405-001 Exam 3 13 April 2018

Complete each problem on a separate page. Absolutely NO electronic devices are

allowed (phones, calculators, headphones, etc.) Show your work and justify every

solution for full credit. You may use any theorems presented in class, but state it

clearly . You have 50 minutes.

Name:

1. (15 points) DO ONLY 3 OUT OF 4: True or False. If true, prove. If false, state why or provide a

counterexample.

(a) There exists a surjective linear map from P2 to M2×2.

(b) The linear map T : M2×2 →M2×2 given by T (A) = At +A is an isomorphism.

(c) The matrix A =

1 −1 0

0 −4 1

−2 −2 1

has a trivial null space.

(d) If A is an invertible n×n matrix and λ is a nonzero eigenvalue for A, then1

λis an eigenvalue for

A−1.

2. (10 points) DO ONLY 2 OUT OF 3: Give an example of the following. Justify all answers!

(a) An isomorphism from M2×2 to R4.

(b) A nonzero 3× 3 matrix with 0 as an eigenvalue.

(c) A nontrivial linear map from R3 to R3 that is not surjective.

3. (15 points) Let T : P2 → P3 be defined by p(x) 7→ xp(x) .

(a) Let B = {1, 1 − x, 1 + x2} be an ordered basis for P2 and B′ = {x3, x2, x, 1} an ordered basis

for P3. Find [T ]B′

B .

(b) Let p = −4x2 − 2x+ 1. Find [p]B .

(c) Find T (p) using part (a) and (b).

4. (10 points) Let A =

1 0 1

3 2 0

0 0 −1

.

(a) Find the characteristic equation of A.

(b) Find all eigenvalues of A.

(c) Chose ONE eigenvalue from (b) and find the corresponding eigenspace.

36Appendix J

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