[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572365-105":53,"doc-detail-572365-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","linear-algebra-cheat-sheet","Linear algebra cheat-sheet","","Linear algebra cheat-sheet by Laurent Lessard provides a compact set of core concepts and formulas for matrices and vectors. It defines matrix multiplication compatibility, transpose rules, inner and outer products, and the inverse including key properties. It summarizes vector norms and norm balls, Cauchy-Schwarz, linear independence, rank relationships, and conditions for invertibility. It also covers solving linear systems, least squares via normal equations, range and nullspace equivalences, rank-nullity, and orthogonal matrices with invariance and Gram-Schmidt.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/linear-algebra-cheat-sheet/572365/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/linear-algebra-cheat-sheet/572365.png","ImageObject",442,249,{"name":88,"@type":89},"Melati","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"When can two matrices be multiplied and how is the product defined?","Question",{"text":109,"@type":110},"Two matrices can be multiplied when the inner dimensions agree (for A of size m×n and B of size n×p). The entry cij of the product is the sum over aikbkj with k matching the shared dimension.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What are the main properties of the matrix inverse?",{"text":114,"@type":110},"If A is square and an inverse B exists with AB=I (or BA=I), the inverse is denoted A^{-1}. The inverse is unique, satisfies (A^{-1})^{-1}=A, relates to transpose as (A^{-1})^T=(A^T)^{-1}, and the inverse of a product follows (AB)^{-1}=B^{-1}A^{-1}.",{"name":116,"@type":107,"acceptedAnswer":117},"How do least-squares solutions relate to the normal equations?",{"text":118,"@type":110},"For overdetermined systems Ax=b with no exact solution, least squares minimizes the 2-norm of the residual: minimize ||Ax-b||_2^2. The minimizer x^ satisfies the normal equations A^T A x^ = A^T b, and yields the closed form x^=(A^T A)^{-1}A^T b when columns of A are linearly independent.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572365,1792021290,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":47},962085570644,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Linear algebra cheat-sheet  \nLaurent Lessard  \nUniversity of Wisconsin{Madison  \nLast updated: October 12, 2016  \nMatrix basics  \nA matrix is an array of numbers. A 2 Rm􀀂n means that: 2 a11 : : : a1n 3  \nA = 6 .. . . . ... 7 (m rows and n columns) 4 am1 : : : amn5  \nTwo matrices can be multiplied if inner dimensions agree:  \nC  \n(m􀀂p)  \n= A B  \n(m􀀂n)(n􀀂p)  \nwhere  \nn  \ncij =X aikbkjk=1  \nTranspose: The transpose operator AT swaps rows and columns. If A 2 Rm􀀂n then AT 2 Rn􀀂m and (AT )ij = Aji .  \n􀀏 (AT )T = A.  \n􀀏 (AB)T = BT AT .  \nMatrix basics (cont'd)  \nVector products. If x ; y 2 Rn are column vectors,  \n􀀏 The inner product is xTy 2 R (a. k.a. dot product)  \n􀀏 The outer product is xyT 2 Rn􀀂n .  \nThese are just ordinary matrix multiplications!  \nInverse. Let A 2 Rn􀀂n (square) . If there exists B 2 Rn􀀂n with AB = I or BA = I (if one holds, then the other holds with the same B ) then B is called the inverse of A, denoted B = A􀀀1 .  \nSome properties of the matrix inverse:  \n􀀏 A􀀀1 is unique if it exists.  \n􀀏 (A􀀀1)􀀀1 = A.  \n􀀏 (A􀀀1)T = (AT )􀀀1 .  \n􀀏 (AB )􀀀1 = B 􀀀1A􀀀1 .  \nVector norms  \nA norm k􀀁k : Rn ! R is a function satisfying the properties:  \n􀀏 kx k = 0 if and only if x = 0 (de􀀌niteness)  \n􀀏 kcx k = jc jkxk for all c 2 R (homogeneity)  \n􀀏 kx + y k 􀀔 kx k + ky k (triangle inequality)  \nCommon examples of norms:  \n􀀏 kx k1 = jx1 j + 􀀁 􀀁 􀀁 + jxn j (the 1-norm)  \n􀀏 kx k2 =px21 + 􀀁 􀀁 􀀁 + x2n  (the 2-norm)  \n􀀏 kx k1 = max 1􀀔i􀀔n jxi j (max-norm)  \nProperties of the 2-norm (Euclidean norm)  \n􀀏 If you see kx k, think kx k2 (it's the default)  \n􀀏 xTx = kx k2  \n1  \nx1  \nnorm ball:  \nfx j kxk = 1g  \n􀀏 xTy 􀀔 kxkky k (Cauchy-Schwarz inequality)  \nLinear independence  \nA set of vectors fx1 ; : : : ; xng 2 Rm is linearly independent if c1x1 + 􀀁 􀀁 􀀁 + cnxn = 0 if and only if c1 = 􀀁 􀀁 􀀁 = cn = 0  \nIf we de􀀌ne the matrix A = 􀀂x1 􀀁 􀀁 􀀁 xn 􀀃 2 Rm􀀂n then the columns of A are linearly independent if  \nAw = 0 if and only if w = 0  \nIf the vectors are not linearly independent, then they are linearly dependent. In this case, at least one of the vectors is redundant (can be expressed as a linear combination of the others) . i.e. there exists a k and real numbers ci such that  \nxk =X cixi  \nik  \nThe rank of a matrix  \nrank(A) = maximum number of linearly independent columns = maximum number of linearly independent rows  \nIf A 2 Rm􀀂n and B 2 Rn􀀂p then  \n􀀏 rank(A) 􀀔 min(m ; n)  \n􀀏 rank(AB) 􀀔 min(rank(A) ; rank(B)) 􀀔 min(m ; n ; p)  \n􀀏 if rank(A) = n then rank(AB) = rank(B)  \n􀀏 if rank(B) = n then rank(AB) = rank(A)  \nSo multiplying by an invertible matrix does not alter the rank.  \nGeneral properties of the matrix rank:  \n􀀏 rank(A + B) 􀀔 rank(A) + rank(B)  \n􀀏 rank(A) = rank(AT ) = rank(ATA) = rank(AAT )  \n􀀏 A 2 Rn􀀂n is invertible if and only if rank(A) = n.  \nLinear equations  \nGiven A 2 Rm􀀂n and b 2 Rm , linear equations take the form  \nAx = b  \nWhere we must solve for x 2 Rn . Three possibilities:  \n􀀏 No solutions. Example: x1 + x2 = 1 and x1 + x2 = 0 .  \n􀀏 Exactly one solution. Example: x1 = 1 and x2 = 0 .  \n􀀏 In􀀌nitely many solutions. Example: x1 + x2 = 0 .  \nTwo common cases:  \n􀀏 Overdetermined: m > n. Typically no solutions. One approach is least-squares; 􀀌nd x to minimize kAx 􀀀 bk2 .  \n􀀏 Underdetermined: m \u003C n. Typically in􀀌nitely many solutions. One approach is regularization; 􀀌nd the solution to Ax = b such that kx k2 is as small as possible.  \nLeast squares  \nWhen the linear equations Ax = b are overdetermined and there is no solution, one approach is to 􀀌nd an x that almost works by minimizing the 2-norm of the residual:  \nminimize kAx 􀀀 bk2 (1)  \nx  \nThis problem always has a solution (not necessarily unique) . x^ minimizes (1) i􀀋 x^ satis􀀌es the normal equations:  \nATA x^ = ATb  \nThe normal equations (and therefore (1)) have a unique solution i􀀋 the columns of A are linearly independent. Then,  \nx^ = (ATA)􀀀1ATb  \nRange and nullspace  \nGiven A 2 Rm􀀂n, we have the de􀀌nitions:  \nRange: R (A) = fAx j x 2 Rng. Note: R (A) 􀀒 Rm  \nNullspace: N (A) = ","cbCaic8KtO2xwnkZ","https://ap.wps.com/l/cbCaic8KtO2xwnkZ","pdf",161563,13,"English","# Matrix basics\n## Transpose\n## Vector products and inverse\n## Vector norms\n# Linear independence and rank\n# Linear equations\n## Least squares\n## Range and nullspace\n# Orthogonal matrices","[{\"question\":\"When can two matrices be multiplied and how is the product defined?\",\"answer\":\"Two matrices can be multiplied when the inner dimensions agree (for A of size m×n and B of size n×p). The entry cij of the product is the sum over aikbkj with k matching the shared dimension.\"},{\"question\":\"What are the main properties of the matrix inverse?\",\"answer\":\"If A is square and an inverse B exists with AB=I (or BA=I), the inverse is denoted A^{-1}. The inverse is unique, satisfies (A^{-1})^{-1}=A, relates to transpose as (A^{-1})^T=(A^T)^{-1}, and the inverse of a product follows (AB)^{-1}=B^{-1}A^{-1}.\"},{\"question\":\"How do least-squares solutions relate to the normal equations?\",\"answer\":\"For overdetermined systems Ax=b with no exact solution, least squares minimizes the 2-norm of the residual: minimize ||Ax-b||_2^2. The minimizer x^ satisfies the normal equations A^T A x^ = A^T b, and yields the closed form x^=(A^T A)^{-1}A^T b when columns of A are linearly independent.\"}]","Linear algebra cheat-sheet | PDF",1791926984]