Midterm MATH 133 PDF

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Math 133 Midterm 1 Exam — Sections 001, 002, 003, 004 Instructors: Dr. Rosalie B´elanger-Rioux ([email protected]) Piotr Przytycki ([email protected]) Instructions. Please read carefully. • Timing. You are allowed to use a maximum of 3.5 (three and a half) consecutive hours (210 consecutive minutes) for this exam (including downloading the questions, answering them, and uploading your answers), and all of this time must be within the following 48 hour window of time: 9:00am Montreal time “UTC/GMT -4 hours” on October 5th, to 9:00am Montreal time “UTC/GMT -4 hours” on October 7th. If you need to take a break during your 3.5hours, then do it quickly because the counter will keep running no matter what! Please plan ahead carefully. Here are some examples: – You may start at 5:00pm Montreal time on October 5th, and you will have until 8:30pm on October 5th to submit your answers online (even though the end of the 48 hour window is on October 7th, since you will have used up your maximum of 3 hours and 30 minutes by 8:30pm). – You may start at 5:00am Montreal time on October 7, and you will have until 8:30am to submit your answers online. – You may start at 6:00am Montreal time on October 7 if you wish, but you will have to submit your answers before 9:00am on that day, even though you will only have used 3 hours of the maximum alloted 3 hours and 30 minutes. – If you live in Hong Kong (UTC +8 hours), your window of time is October 5 9:00pm local time to October 7 9pm local time. – Pace yourself, and do not spend too much time on any one problem. The exam has been written to take 1.5 hours to complete, but it is McGill policy to give you all double time to account for possible disability issues, with an extra thirty minutes to allow for connection and uploading issues, which is why you are allowed 3 hours and a half. • Terms and conditions. – You may consult any textbook or course notes or videos you like (whether paper or online). You may not copy answers from those sources, but you may look through those sources to remind yourself of the major definitions and results from class and to look through practice problems. – You may not use a calculator or any software that can perform calculations, not even “just” to check your answer. – You may not use the help of any person or other resource while completing this exam; including but not limited to a friend, teacher, tutor, any online discussion forum or other form of a help forum, etc. Office hours will be cancelled during the 48 hour window of the exam. The discussion forum on our myCourses website will be locked, so you will be able to consult it but unable to post to it. Tutorials on Tuesday will still happen, but will cover material from the week before and will not address questions you may have about the exam. – Please write legibly and dark enough. Please check your work is still legible after having taken pictures or scans of it (what you will upload to Crowdmark). – This exam is the intellectual property of McGill University and the course instructors. Please do not share any part of it in any way with anyone. See next page for more instructions. 1

• Academic integrity. McGill University and your course staff value academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (approved by Senate on 29 January 2003, see www.mcgill.ca/students/srr/honest/ for more information). Write out and sign the following statement and make that your submission for the last question on this exam, if it applies to you: I have neither given nor received unauthorized aid on this exam and I agree to adhere to the Instructions that govern this exam. • Obtaining the exam and submitting your answers. Click on the “Crowdmark” tab of our top navbar in myCourses. You should see a button that says “sign in through myCourses”, click on it. You should see a list of courses (ours might be the only one). Click on Math 133, and you will see a list of “assessments.” We will use Crowdmark for midterms and the final exam, and a “trial” assessment so you can test the platform before our first midterm. So you may only see the trial assesment when you first sign in. – At the exam start time (9am on October 5, Montreal time “UTC/GMT -4 hours”), you will automatically receive an email from Crowdmark giving you access to the exam. You may decide when you “start” the exam (and thus when you are allowed to first look at the questions). From the moment you “start”, you will have 3.5 hours to read the questions, answer each question, and submit your answer to each question on Crowdmark. – You have three options for submission: (1) write out your answers using paper and pen or pencil, take pictures or scans of each page and upload them on Crowdmark; (2) write out your answers electronically using a tablet or other drawing device, and upload your answers as pictures or pdfs to Crowdmark; (3) type out your answers on software such as a word or LaTeX processor, convert to pdf and submit to Crowdmark. – No late work will be accepted. You may submit answers for each question at any time during your 3.5 hours inside the allowed window of time. You may also resubmit: only the latest submission for each question will be seen by the grader. So submit early, and resubmit if necessary. • Questions that arise during the exam. If you find an exam question to be ambiguous or unclear, email your instructor. Of course it’s likely the response will be “sorry, I can’t answer that.” Do not ask questions to anyone else. • Exam format. This exam paper has a total of 60 points and consists of 7 questions (plus one question for you to write out an academic integrity statement). Show and justify each step in the solution, and simplify final answers. Answers with no explanation will receive no credit, even if they are correct. Remember that an objective of this course is for you to be able to clearly explain your reasoning to others.

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1. (Worth no points but absolutely mandatory.) McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (approved by Senate on 29 January 2003, see www.mcgill.ca/students/srr/honest/ for more information). Write out and sign the following statement and make that your submission for this first question of the exam: I have neither given nor received unauthorized aid on this exam and I agree to adhere to the Instructions above that govern this exam. 2. (8 points) True or false? If true, give a concise explanation; if false, give a counterexample. (a) A system of 3 linear equations in 2 unknowns (i.e. variable) has at most one solution. (b) If span(~v1 , ~v2 , ~v3 , ~v4 ) is a subspace of dimension 3, then ~v1 , ~v2 , ~v3 are linearly independent. 3. (8 points) Consider the system of equations 1x + 3y + 2z = 4 3x + 7y + 8z = 10 a) Find all solutions to this system of equations. Show your work. b) Find values for the scalars a, b,c, d(those values should all be non-zero), such that the system 11 below has the unique solution −1. Explain or show your work. −2 1x + 3y + 2z = 4 3x + 7y + 8z = 10 ax + by + cz = d 4. (8 points)      0 0 k+2 (a) For which value(s) of k ∈ R is the vector  1  a linear combination of  −1 and  2? 3 0 0 Justify your answer.       2 1 1 (b) For which value(s) of k ∈ R is the vector  k  a linear combination of  −2 and  −4? −5 −3 −5 Justify your answer. 

5. (10 points) Consider the following set:      x W =  y  ∈ R3 , such that − x + y − z = 0   z (a) Justify the assertion that W is a subspace of R3 . (b) Give a basis for W . (c) Give the dimension of W . Justify every part of your answer.

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6. (8 points) True or false? Explain. (a) If T : R3 → R2 is a linear transformation and ~c is a vector in R2 , then the equation T (~ x) = ~c has either no solution or infinitely many solutions.   3 −2 are the (b) The vectors u ~ that satisty the equation A~ u = 133~ u with the matrix A = 0 8   −130 −135 0 . solutions of the following system, written in an augmented matrix: −133 −125 0 7. (10 points) The given transformations T : R2 → R2 below are all linear. (a) Let T be the reflection across the x axis. What is T (~ e1 )? What is T (e~2 )? What is the matrix A satisfying T (~v ) = A~v for all ~v ∈ R2 ? Justify your answers. (b) Let T be the transformation such that         10 −2 −15 1 = and T = . T −3 5 1 25 What is T (~e1 )? What is T (~e2 )? What is the matrix A satisfying T (~v) = A~v for all ~v ∈ R2 ? Justify your answers. 8. (8 points) Consider the matrix  −2 0 3 1 . −4 4

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1 A =  −2 0

  3 ~ (a) Let b = A~v, where ~v = 2 . Find ~b. Show your work. 0

(b) Find a vector x ~ ∈ R3 which is not a solution to the system A~ x = ~b (same ~b as in part a). It is fine to guess, but you still need to explain how you know your chosen x ~ is not a solution.

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