[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-513707-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-513707-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","linear-algebra-matrix-analysis-and-least-squares","Linear Algebra - Matrix Analysis and Least Squares","","This document provides a technical overview of fundamental concepts in linear algebra, focusing on matrix spaces, projections, and least squares problems. It explores the relationships between the four fundamental subspaces—column space R(A), row space R(AT), null space N(A), and left null space N(AT)—and illustrates their geometric interactions. The material covers overdetermined systems, the normal equations for solving least squares (ATAx=ATb), and the application of Singular Value Decomposition (SVD) for matrix factorization. The content serves as a review of how linear transformations map between these spaces and provides insights into minimizing the residual error ||b-p||^2 in regression problems. The analysis is supported by illustrative diagrams depicting orthogonal complements and projections, facilitating a deeper understanding of matrix rank and system solvability.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/linear-algebra-matrix-analysis-and-least-squares/513707/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/linear-algebra-matrix-analysis-and-least-squares/513707.png","ImageObject",300,407,{"name":42,"@type":43},"Jordan Avery","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-27","2026-10-06",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the four fundamental subspaces associated with a matrix A?","Question",{"text":63,"@type":64},"The four fundamental subspaces are the column space R(A), the null space N(A), the row space R(AT), and the left null space N(AT).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are least squares solutions obtained for an overdetermined system Ax=b?",{"text":68,"@type":64},"The least squares solution is found by solving the normal equations, ATAx = ATb, which minimizes the squared norm of the residual vector ||b - Ax||^2.",{"name":70,"@type":61,"acceptedAnswer":71},"What is the geometric interpretation of the residual vector in a least squares problem?",{"text":72,"@type":64},"The residual vector (b - p), where p is the projection of b onto the column space of A, is orthogonal to the column space of A and resides in the left null space N(AT).","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},513707,1792233220,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":96,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":144,"preview_page_count":4},1099523882367,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","√3/N6Ax=6~11120peDN③个1000peRCA)preanA×=P21eeSDNIb-√:的少的e心n/0.09PeRCA)oeDNn|Ax-bll²2八o30~1b早(×ATA×=ATbx=(ATA)'A'b5(ATA)一5 1oJrank(A'A)=cank(A)④ecs1erank(A)=nkp√ATA:Rn→1Rnoy65N2seo~pr Z13A 6一131neJ1)③◎rank(ATA)sn  \n(rank(A)=n  \n②  \n:элээЭp⊥b-P3D户eaDNPL-⑤A(1人e pmNle wll space了yre(bP)b-peN(A?e? o=A×:peoN今A(b-Ax)=0ieS今↓①20ATA.yfe 7~一lcjf 》3J间GJK1.~3七,b₁)3/D10003见6N- m(七m, bm)旦一一18)21yew火1(1)(2,327.150.3o=×,+X₂七:121122七m  \na③IN13.Byle0g1 0395220N93\"DkJe一、2007105n;(t)=n;(o)·e入:t0“DN儿~i/ONP'7⁹”n00NO12人”.6N13n七)=2n(o)e7t七,tz…,tmpynen(+)e3'3NL入L3c3?n,(o),…,n(o)1c3~12CNLinearleaotU9NeSqwaresk2D5DCeUnderdetermined人28~55ALe人en>m  \n③  \n一人Q10  \nR  \nR(AT)R(A)PN(A).Pc1D,RCA)6 bNep小GN(.Ax+=p.见9x↑eR(A7PeoDN(2ASVD- SinularValwe①ecompositkor李明|刘敏|张涵mxnL0A分3GN6SvDCOeNA=UDyT22一03n×nAAT6~3y 021G9-4“10c”山xATAVJP少1/e e·Jc 一801厂).Cp○2472。Pee p力一110k-D(AATLe yx 0PDe)AT4 瓦 corroo-!4e 3N6②390ycl32J'2302¥x山P'230N分3 白230P\"9》/y3  \n⑥  \n心么っ/分3CN一少D klkD32CNl3D/'Ne广'c)'qEC  二〇4(Gh②=”10cNe)2BW×r31A3GN2106四3Ac=Ip/C在Q少0JCBA=Inp/sA e /0.ApIclB= BCAc)BAC二A)c一少～分◎pc人vele0》2～6⑨\"3y'?cJ131NUe中olcpQTQ=InksQe～e01cQ⁷≠Ims/y'比4P/SI","cbCaijPsNzhyK0se","https://ap.wps.com/l/cbCaijPsNzhyK0se","pdf",2228490,"English","# Fundamental Subspaces of a Matrix\n## Geometric Relationships and Projections\n# Solving Least Squares Problems\n## The Normal Equations\n## Overdetermined Systems\n# Singular Value Decomposition (SVD)\n## Matrix Factorization and Applications","[{\"question\":\"What are the four fundamental subspaces associated with a matrix A?\",\"answer\":\"The four fundamental subspaces are the column space R(A), the null space N(A), the row space R(AT), and the left null space N(AT).\"},{\"question\":\"How are least squares solutions obtained for an overdetermined system Ax=b?\",\"answer\":\"The least squares solution is found by solving the normal equations, ATAx = ATb, which minimizes the squared norm of the residual vector ||b - Ax||^2.\"},{\"question\":\"What is the geometric interpretation of the residual vector in a least squares problem?\",\"answer\":\"The residual vector (b - p), where p is the projection of b onto the column space of A, is orthogonal to the column space of A and resides in the left null space N(AT).\"}]","Linear Algebra - Matrix Analysis and Least Squares | PDF",1791289308,13]