[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-455905-105":59,"doc-detail-455905-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","linear-algebra-practice-final-spring-2019","Linear Algebra Practice Final - Spring 2019","","Linear Algebra practice final for Spring 2019, focusing on core matrix and vector analysis. Problems require computing AAT and assessing diagonalizability via theorems or explicit demonstrations, evaluating determinants and linear independence of transformed matrices, and determining eigenvalues, geometric multiplicities, and diagonalizability for 2x2 matrices. Additional tasks cover similarity between matrices, coordinates and matrix representations for projections and orthogonal decompositions in R2, eigenanalysis and orthogonal diagonalizability, computing e^A using diagonalization, verifying Hermitian structure, and proving orthogonality preserves the dot product for matrices in O(n).",{"@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/linear-algebra-practice-final-spring-2019/455905/",{"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/linear-algebra-practice-final-spring-2019/455905.png","ImageObject",300,407,{"name":92,"@type":93},"ahduag","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you determine whether AAT is diagonalizable in Problem 1?","Question",{"text":113,"@type":114},"Compute AAT and justify diagonalizability using a clear demonstration or an appropriate diagonalization theorem referenced in the solution.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is required to analyze ATA in Problem 2?",{"text":118,"@type":114},"First find det(ATA). Then use that result to decide whether the columns of ATA are linearly independent.",{"name":120,"@type":111,"acceptedAnswer":121},"What methods are used in Problem 5 to compute e^A?",{"text":122,"@type":114},"Find eigenvalues and eigenvectors to form a diagonal matrix D and a similarity factorization A = PDP^{-1}. Then use the relationship e^A = P e^D P^{-1} to compute the matrix exponential.","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},455905,1790917441,{"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":146},3985747870414,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Linear Algebra Practice Final  \nSpring 2019  \n1. (10 points) Let  \nA = 􀀒 10 􀀀 31 02􀀓  \nCompute AAT and determine if it is diagonalizable. Justify your answer either with a clear demonstration or with reference to a relevant theorem.  \n2. Using the same matrix A as in problem (1),  \nA = 􀀒 10  \n3  \n􀀀1  \n02􀀓  \nanswer the following questions:  \n(a) (5 points) What is det(ATA)?  \n(b) (5 points) Use the above result to determine if the columns of ATA are linearly independent.  \n3. Consider the matrices A = 􀀒 20 12􀀓 and B = 􀀒 20 02􀀓 .  \n(a) (5 points) Find the eigenvalues of each matrix.  \n(b) (5 points) Determine the geometric multiplicities of each eigenvalue, for each matrix. Which of A and B is diagonalizable?  \n(c) (5 points) Determine whether the two matrices are similar, i.e. whether B = PAP 􀀀1 for some P 2 GL(2 ; R) .  \n4. Consider the vector v = 􀀒 52􀀓 in R2 .  \n(a) (5 points) Determine [v]􀀌 if 􀀌 = 􀀒􀀒 12􀀓 ; 􀀒34􀀓􀀓 .  \n(b) (5 points) Find the standard representation of the projection 􀀙 w onto the vector w = R 􀀙=2v orthogonal to v.  \n(c) (5 points) Find the standard representation of the re􀀍ection R ` across ` = span(v) .  \n5. Let  \nA = 􀀒 23 􀀀 36􀀓  \n(a) (5 points) Find the eigenvalues of A.  \n(b) (5 points) Find corresponding eigenvectors of A.  \n(c) (5 points) Is A orthogonally diagonalizable? Justify your answer with a clear demonstration or with reference to a relevant theorem.  \n(d) (5 points) Let  \nD = [A]􀀌 = 􀀒 􀀕 10 0􀀕 2􀀓  \nbe the diagonal matrix with the eigenvalues of A, 􀀕 1 and 􀀕 2 , you got in part (b), whose corresponding eigenvectors v 1 and v2 you got in part (c) . Use the fact that A = PDP 􀀀1, and therefore eA = eP DP 􀀀1 = PeDP􀀀1, to compute eA .  \nHere, of course, eA  P ~~1~~n! An.  \n6. (5 points) Show that A = 􀀒 2  i 2  i􀀓 is Hermitian.  \n7. (5 points) Let A 2 O(n) be an n 􀀂 n orthogonal matrix. Show that  \n(Ax) 􀀁 (Ay) = x 􀀁 y  \nfor all x ; y 2 Rn. Hint: the dot product x 􀀁 y can be computed in terms of matrices,  \n0 y 11  \nx 􀀁 y = 􀀀x1 􀀁 􀀁 􀀁 xn􀀁 B .. C = xTy  \n@y.nA","cbCaieIQRJD8mRLL","https://ap.wps.com/l/cbCaieIQRJD8mRLL","pdf",221802,17,"English","# Linear Algebra Practice Final\n## Problem 1\n## Problem 2\n## Problem 3\n## Problem 4\n## Problem 5\n## Problem 6\n## Problem 7","[{\"question\":\"How do you determine whether AAT is diagonalizable in Problem 1?\",\"answer\":\"Compute AAT and justify diagonalizability using a clear demonstration or an appropriate diagonalization theorem referenced in the solution.\"},{\"question\":\"What is required to analyze ATA in Problem 2?\",\"answer\":\"First find det(ATA). Then use that result to decide whether the columns of ATA are linearly independent.\"},{\"question\":\"What methods are used in Problem 5 to compute e^A?\",\"answer\":\"Find eigenvalues and eigenvectors to form a diagonal matrix D and a similarity factorization A = PDP^{-1}. Then use the relationship e^A = P e^D P^{-1} to compute the matrix exponential.\"}]","Linear Algebra Practice Final - Spring 2019 | PDF",1790744637,43]