Math 415 - Final ExamFriday, December 12, 2014Circle your section:Philipp Hieronymi 2pm 3pmArmin Straub 9am 11amName:NetID:UIN:To be completed by the grader:0 1 2 3 4 5 MCP/1 /? /? /? /? /? /? /??Good luck!Instructions• No notes, personal aids or calculators are permitted.• This exam consists of ? pages. Take a moment to make sure you have all pages.• You have 180 minutes.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page (make it clear if you do).• Explain your work! Little or no points will be given for a correct answer with noexplanation of how you got it.• In particular, you have to write down all row operations for full credit.1Important Note• The collection of problems below is not representative of the final exam!• The first three problems cover the material since the third midterm exam, andproblems on the final exam on these topics will be of similar nature.• Problems 4 and 5 are a good start to review the material we covered earlier; however,on the exam itself you should expect questions of the kind that we had on theprevious midterms.• In other words, to prepare for the final, you need to also prepare our pastmidterm exams and practice exams.• In particular, a basic understanding of Fourier series or the ability to work withspaces of polynomials are expected.Problem 1. Find a solution to the initial value problem (that is, differential equation plusinitial condition)ddtu =1 1 01 0 10 1 1u, u(0) =210.Simplify your solution as far as possible.Problem 2. The processors of a supercomputer are inspected weekly in order to determinetheir condition. The condition of a processor can either be perfect, good, reasonable or bad.A perfect processor is still perfect after one week with probability 0.7, with probability 0.2the state is good, and with probability 0.1 it is reasonable. A processor in good conditionsis still good after one week with probability 0.6, reasonable with probability 0.2, and badwith probability 0.2. A processor in reasonable condition is still reasonable after one weekwith probability 0.5 and bad with probability 0.5. A bad processor must be repaired. Thereparation takes one week, after which the processor is again in perfect condition.In the steady state, what is percentage of processors in perfect condition?Problem 3. Determine the PageRank vector for the following system of webpages, and rankthe webpages accordingly.AB CDEF2Problem 4. Consider A =1 1 01 0 10 1 1.(a) Find bases for Nul(A) and Col(A).(b) Determine the LU decomposition of A.(c) Determine the inverse of A.(d) What is the determinant of A?(e) Determine the QR decomposition of A.(f) Determine the eigenvalues of A and find bases for the eigenspaces.(g) Diagonalize A.Problem 5. Consider A =1 11 00 1.(a) Find orthogonal bases for all four fundamental subspaces.(b) Determine the projection matrices corresponding to orthogonal projection onto Col(A)and Col(AT).(c) Consider the linear function T : R2→ R3, which maps x to Ax.• Determine the matrix which represents T with respect to the standard bases of R2and R3.• Determine the matrix which represents T with respect to the basis10,11for R2,and100,110,111for R3.(d) Find the least squares solution to Ax =111.3MULTIPLE CHOICE(? questions, 2 points each)Instructions for multiple choice questions• No reason needs to be given. There is always exactly one correct answer.• Enter your answer on the scantron sheet that is included with your exam.In addition, on your exam paper, circle the choices you made on the scantron sheet.• Use a number 2 pencil to shade the bubbles completely and darkly.• Do NOT cross out your mistakes, but rather erase them thoroughly before enteringanother answer.• Before beginning, please code in your name, UIN, and netid in the appropriateplaces. In the ‘Section’ field on the scantron, please enter000 if Armin Straub is your instructor,001 if Philipp Hieronymi is your instructor.The actual exam will have multiple choice questions here.The midterm exams as well as the practice exams have plenty of problems that you can (andshould) look at again. Below are the short problems and multiple choice questions from theconflict exam of our midterms.Shorts 1. Let A =1 00 11 0. Compute ATA.Shorts 2. Let A be a matrix such that, for everyxyzin R3, Axyz=−zx + y2x + z.Then, what is A?Shorts 3. Let C be a 3 ×3 matrix such that C has three pivot columns, and let d be a vectorin R3. Is it true that, if the equation Cx = d has a solution, then it has infinitely manysolutions?(a) True.(b) False.(c) Unable to determine.4Shorts 4. LetA =a − 1 aa a − 1.For which choice(s) of a is the matrix A not invertible?Shorts 5. Write down a 3 ×3-matrix that is not the zero matrix (ie the matrix whose entriesare all zero) and is not invertible.Shorts 6. Let W1be the set of all 2 × 2-matrices A such that A is invertible, and let W2bethe set of all 2 × 2-matrices A such that AT= −A. Are these sets subspaces of the vectorspace of all 2 × 2-matrices?(a) Both W1and W2are subspaces.(b) Only W1is a subspace.(c) Only W2is a subspace.(d) Neither W1nor W2are subspaces.Shorts 7. Let W = span100,010,001. Which of the following is true?(a) W is empty.(b) W is a line.(c) W is a plane.(d) W is all of R3.Shorts 8. Let H be a subspace of R6with basis {b1, b2, b3, b4, b5}. What is the dimension ofH?5Shorts 9. Which of the following collections of vectors is linearly independent?(a) {101,01−1,321}(b) {1−10,01−1,−101}(c) {123,111,−101}(d) {110,011,101}Shorts 10. Let A be an 4 ×5 matrix of rank 2. Is it possible to find two linearly independentvectors that are orthogonal to the null space of A? Is it possible to find two linearly independentvectors that are orthogonal to the left null space of A?(a) Possible for both.(b) Possible only for the column space.(c) Possible only for the row space.(d) Not possible in either case.(e) Not enough information to decide.Shorts 11. Let P2be the vector space of all polynomials of degree up to 2, and let T : P2→ P2be the linear transformation defined
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