[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-394235-105":59,"doc-detail-394235-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","math-218-practice-midterm-examination-3","MATH 218 - Practice Midterm Examination 3","","MATH 218 Section 3 provides instructions and a set of practice midterm problems to mirror the real exam’s format, length, and difficulty. The exam is closed book for a 75-minute session, with guidance on pacing and showing work for full credit. Problems cover eigenvalues and diagonalization of matrices, orthogonal similarity, differential equation systems with initial conditions, projection matrices and their eigenspaces, positive definiteness for matrix sums, singular value decomposition and related vectors, and constructing example matrices under specific algebraic and spectral constraints.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/math-218-practice-midterm-examination-3/394235/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/math-218-practice-midterm-examination-3/394235.png","ImageObject",300,407,{"name":92,"@type":93},"Eliana","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the exam rules and allowed resources?","Question",{"text":113,"@type":114},"The practice exam is closed book. You may use a calculator for arithmetic, but no other technology is allowed.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How should students approach the problems for full credit?",{"text":118,"@type":114},"Show your work so the reasoning is clear. Avoid spending too much time on any single problem and read through all problems first.",{"name":120,"@type":111,"acceptedAnswer":121},"Which topics are included in the practice midterm problems?",{"text":122,"@type":114},"The problems include eigenvalues and diagonalization, orthogonal similarity, solving differential equation systems, eigenvalues of projection matrices, positive definiteness of matrix sums, singular value decomposition, and constructing matrices meeting specific spectral and determinant conditions.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},394235,1790528174,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":41,"preview_page_count":4},4398048949847,"https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267","MATH 218 SECTION 3  \nPRACTICE MIDTERM EXAMINATION 3  \n\n| Name |  | Duke UniqueID |  |\n| --- | --- | --- | --- |\n\nPlease read all instructions carefully before beginning.  \n􀀏 Do not open this test booklet until you are directed to do so.  \n􀀏 You have 75 minutes to complete this exam.  \n􀀏 If you finish early, go back and check your work.  \n􀀏 This exam is closed book.  \n􀀏 You may use a calculator to do arithmetic, but you should not need one. No other technology is allowed.  \n􀀏 For full credit you must show your work so that your reasoning is clear.  \n􀀏 Do not spend too much time on any one problem. Read them all through first and attack them in an order that allows you to make the most progress.  \n􀀏 Good luck!  \nThis is a practice exam. It is meant to be similar in format, length, and difficulty to the real exam. It is not meant as a comprehensive list of study problems. I recommend completing the practice exam in 75 minutes, without notes or distractions.  \n[This page intentionally left blank]  \nProblem 1. [20 points]  \nConsider the matrix  \nA = 􀂁 23 3636 2 􀂋 .  \na) Find the eigenvalues of this matrix. (One of the eigenvalues is 􀀀25.)  \nb) Find an orthogonal matrix Q and a diagonal matrix 􀀃 such that A = Q􀀃QT .  \nc) Consider the following system of differential equations with initial values:  \n~~d ~~x~~d ~~t = 23x + 36y x (0) = 7 d~~d~~yt = 36x + 2y y (0) = 􀀀1.  \nFind the solutions x( t ) and y ( t ) .  \nd) Find a formula forAn in terms of n.  \ne) Find the vector v = 􀀀 xy􀀁 maximizing j vj, subject to the constraint 23x 2 + 72xy + 2y 2 = 1.  \n[Scratch work for problem 1]  \nProblem 2. [10 points]  \nLet V be a subspace of R4 ; assume that V  f0g and V  R4. Let P be the matrix for the projection onto V .  \na) Show that the eigenvalues of P are 0 and 1 .  \nb) Which of the four fundamental subspaces is equal to the 0-eigenspace? Which of the four fundamental subspaces is equal to the 1-eigenspace?  \nc) Why is P diagonalizable? What diagonal matrix is it similar to?  \n[Scratch work for problem 2]  \nProblem 3. [5 points]  \nShow that if S and T are positive definite, then so is S + T . [Hint: knowing that the eigenvalues are positive is not helpful here.]  \nProblem 4. [15 points]  \nConsider the matrix  \nA = 􀂁 13 13 􀂋 .  \nThis question is about the singular value decomposition A = U􀀆VT . Let U = 􀀀 u1 u2 􀀁 and V = 􀀀 v1 v2 􀀁 .  \na) Find u1 and v1 without computingATA orAAT . [Hint: in which spaces do these live?]  \nb) Using your answers to (a) or otherwise, compute 􀀛 1.  \nc) Write down the singular value decomposition A = U􀀆VT ofA.  \n[Scratch work for problems 3 and 4]  \nProblem 5. [15 points]  \nGive examples of matrices with the following properties. If no such matrix exists, write“not possible”. All matrices in this problem have real entries.  \na) A 3 􀀂 2 matrix with singular values 2 and 1 .  \nb) A 2 􀀂 2 matrix that is invertible but not diagonalizable.  \nc) A 3 􀀂 3 symmetric matrix that is positive semidefinite but not positive definite.  \nd) A 3 􀀂 3 symmetric matrix that is not diagonalizable.  \ne) A 3 􀀂 3 matrix whose entries are all either 1 or 􀀀1, with determinant 4 .[Hint: think in terms of maximizing volumes.]  \n[Scratch work for problem 5]","cbCaidiQ6HIi22kL","https://ap.wps.com/l/cbCaidiQ6HIi22kL","pdf",63539,12,"English","# Instructions\n# Problem 1: Eigenvalues, diagonalization, differential equations, sequences, optimization\n# Problem 2: Projection matrix eigenvalues and eigenspaces\n# Problem 3: Positive definiteness of S + T\n# Problem 4: Singular value decomposition and singular vectors\n# Problem 5: Constructing matrices with specified properties","[{\"question\":\"What are the exam rules and allowed resources?\",\"answer\":\"The practice exam is closed book. You may use a calculator for arithmetic, but no other technology is allowed.\"},{\"question\":\"How should students approach the problems for full credit?\",\"answer\":\"Show your work so the reasoning is clear. Avoid spending too much time on any single problem and read through all problems first.\"},{\"question\":\"Which topics are included in the practice midterm problems?\",\"answer\":\"The problems include eigenvalues and diagonalization, orthogonal similarity, solving differential equation systems, eigenvalues of projection matrices, positive definiteness of matrix sums, singular value decomposition, and constructing matrices meeting specific spectral and determinant conditions.\"}]","MATH 218 - Practice Midterm Examination 3 | PDF",1790400794]