[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572366-105":53,"doc-detail-572366-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-january-2024","Linear Algebra Cheat Sheet - January 2024","","Linear Algebra Cheat Sheet by Yiping Lu (January 2024) consolidates core concepts for vectors, matrices, linear systems, operations, elimination, and inversion. It summarizes vector algebra including addition, scalar multiplication, dot product, length, angle, and span. It defines matrix dimensions and shows multiplication via row/column representations, then explains linear dependence/independence and solution cases for Ax=b. It further lists matrix operation identities and Gaussian elimination using augmented matrices and pivot-based transformations.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/linear-algebra-cheat-sheet-january-2024/572366/",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-january-2024/572366.png","ImageObject",442,249,{"name":88,"@type":89},"Bintang","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How are vector dot product, length, and angle between two vectors computed in the cheat sheet?","Question",{"text":109,"@type":110},"It gives the dot product as Ax·Bx + Ay·By + Az·Bz, length as ‖A‖ = √(Ax^2 + Ay^2 + Az^2), and the angle as θ = arccos((A·B)/(‖A‖‖B‖)).","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What conditions determine whether the linear system Ax=b has no solution, a unique solution, or infinitely many solutions?",{"text":114,"@type":110},"It states Ax=b has no solution unless b lies in the span of the column vectors of A; in the unique-solution case, A must be square and its columns must be linearly independent, equivalently A has an inverse and x = A^{-1}b. Infinite solutions occur when the system is consistent but not uniquely determined (dependent columns / reduced row echelon form with free variables).",{"name":116,"@type":107,"acceptedAnswer":117},"How does the cheat sheet describe Gaussian elimination using augmented matrices?",{"text":118,"@type":110},"It defines the augmented matrix [A|b], then uses row operations (scaling a row, replacing a row by a multiple plus another row, and swapping rows) to transform [A|b] toward [I|x]. It also explains elimination and permutation matrices that represent these row transformations.","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},572366,1792033498,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"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":79},962085564381,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Linear Algebra Cheat Sheet  \nYiping Lu  \nJanuary 2024  \n1 Vector and Matrix  \nVectors  \n⃗a = 􀀲􀀴AAAxyz􀀳􀀵 , ⃗b = 􀀲􀀴BBBxyz􀀳􀀵 Vector Addition:⃗a + ⃗b = 􀀲􀀴AAAxyz  BBBxyz􀀳􀀵 Vector Scalar Multiplication:c⃗A = 􀀲􀀴cccAAAxyz􀀳􀀵  \nDot Product: ⃗A · ⃗B = Ax · Bx + Ay · By + Az · Bz (Dot Product is a linear combination, ⃗A · ⃗B = ⃗A⊤ ⃗B in lecture 3)  \nLength: ∥⃗A∥ = qA2x + A2y + A2z  \nAngle Between Two Vectors: θ = arccos 􀀐 ~~ ~~∥⃗A··⃗B∥ 􀀑  \nSpan by ChatGPT The \"span\" of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it’s the space formed by stretching, shrinking, or combining the vectors using scalar multiplication and addition. Geometrically, the span of vectors in 2D or 3D space forms a plane ora subspace. In higher-dimensional spaces, it forms a hyperplane or a subspace.  \nMatrix Examples see [https://2prime.github.io/files/linear/matrixvector](https://2prime.github.io/files/linear/matrixvector). pdf  \nMatrix A ∈ Rm ×n means m rows and n colums matrix. Matrix A can be multiply with a vector v in Rn  \n• Row Representation  \nA = 􀀶􀀲 rr· 􀀳􀀷 , ri ∈ Rn , Av = 􀀶􀀲 r1r2 ···· vv 􀀳􀀷 = 􀀲􀀶􀀶 rr··vv􀀳􀀷  \n􀀴rm ⊤􀀵 􀀴rm · v􀀵 􀀴r⊤mv􀀵  \n• Column Representation  \n􀀲 x 1 􀀳  \nA = 􀀂⃗v1 ,⃗v2 , ··· ,⃗vn􀀃 ,⃗vi ∈ Rm ,⃗x = 􀀶 x·2· 􀀷 A⃗x = |x 1 |⃗v1 +x2 ⃗v2 + ··· + xn ⃗vn  \n􀀴xn 􀀵 sc{azla}r ve{zcto}r  \nLinear Systems For a linear system  \n􀀸􀀾 a 11 x 1 + a 12 x2 + ··· + a 1nxn = b 1  \n􀀾  \n􀀼􀀾 a·2x 1 + a22 x2 + ··· + a2nxn = b2 (1)  \n􀀾  \n􀀾  \n􀀾  \n􀀺 am1x 1 + am2x2 + ··· + amnxn = bm  \nWe can understand it as  \n• a linear combiniation problem: if vector ⃗b = 􀀲􀀶 b1b2 􀀳􀀷 lies in the span of  \n􀀴···bm􀀵  \n􀀶􀀲 a11a21 􀀳􀀷 􀀲􀀶 a12a22 􀀳􀀷 􀀲􀀶 a1na2n 􀀳􀀷|􀀶􀀴a··m·1􀀵􀀷 , 􀀴􀀶 a··m·2􀀵􀀷 , ··· , 􀀴􀀶 a···mn􀀵􀀷  \n}  \n{z  \ncolumn vecrtors of matrix A  \n􀀲 x 1 􀀳 􀀲 b 1 􀀳  \n• a matrix equation A⃗x = ⃗b, ⃗x = 􀀶 x·2· 􀀷 (n unknown), ⃗b = 􀀶 b·2· 􀀷 (right hand  \n􀀴xn 􀀵 􀀴bm 􀀵  \nside of m equations)  \nA = 􀀲􀀶 a11a21 a12a22 · · · a1na2n 􀀳􀀷 ∈ Rm ×n  \n􀀴a· ·m·1 a· ·m·2 · · · a···mn􀀵 Linear depedent and linear independent:  \n• ⃗v1 ,⃗v2 , ··· ,⃗vn are linear depedent if there exists c 1 , ··· , cn which are not all zero such that c 1 ⃗v1 + c2 ⃗v2 ··· + cn ⃗vn = ⃗0  \n• ⃗v1 ,⃗v2 , ··· ,⃗vn are linear indepedent if only c 1 = c2 = ··· = cn = 0 can make c 1 ⃗v1 + c2 ⃗v2 ··· + cn ⃗vn = ⃗0  \nLinear system Ax = b (NOT Required! Only need to know what’s the inverse and not all matrix have an inverse)  \n• A don’t have an inverse:  \n– have solution if and only if b ∈ span{⃗v1 , ··· ,⃗vn } where ⃗v1 , ··· ,⃗vn are column vectors of matrix A  \n– In this case A is not invertible, the linear system have 0 solutions or infinite solutions.  \n• have a unique solution if and only if matrix A have an inverse matrix A −1 . The unique solution is x = A −1b. In this case, (important) A must be a square matrix and ⃗v1 , ··· ,⃗vn are linear independent.  \n2 Matrix Operations  \nA ∈ Rm×n, B ∈ Rn×k then C = AB ∈ Rm×k .  \n􀀲⃗r⊤1⃗v1 ⃗r⊤1⃗v2 ··· ⃗r⊤1⃗vk 􀀳  \n􀀶􀀲⃗r⊤1􀀳􀀷 􀀶 {a ⃗r⊤2⃗v2 ··· ⃗r⊤2⃗vk 􀀷  \nAB = 􀀶 · 􀀷 􀀂⃗v1 , ··· , ⃗vk 􀀃 = 􀀶 · · · 􀀷 (2)  \n􀀶􀀴⃗r⊤m· 􀀵􀀷 􀀶􀀴⃗r⊤m··⃗v1 ⃗r⊤m··⃗v2 ··· ⃗r⊤m··⃗vk􀀵􀀷  \n• (AB)C = A (BC),A (B + C) = AB + AC, c(AB) = A(cB)  \n• AB  BA  \n• A [⃗v1 , ··· ,⃗vk ] = [A⃗v1 , ··· , A⃗vk]  \n• 􀀲􀀶 r⃗r⃗􀀳􀀷 A = 􀀲􀀶 r⃗r⃗AA􀀳􀀷  \n􀀴·􀀵 􀀴r⃗·⊤k· A· 􀀵  \n• AIn = A,ImA = A  \n• For a square matrix A, AA −1 = A −1A = I  \n• (AB)⊤ = B ⊤ A⊤  \n• A upper triangular matrix times a upper triangular matrix is a upper triangular matrix. A Lower triangular matrix times a lower triangular matrix is a upper triangular matrix.  \n3 Elimination  \nThe Agumentation matrix of linear system Ax = b is [A|b] . Opertations you can choose to simplify the system:  \n• Scalar multiply a row  \n• Replace row (j) by ∗ ∗ row(i) + row(j) .  \n• switch two row  \nFirst Way Eliminate to Upper Triangular Form and then solve the upper triangular form.  \nSecond Way (Easier to implement) These transformations provides equivalent systems.  \nSolving the equation equivalent to transform augmented matrix [A|b] to [I|x], we do it column by column  \nExample:  \n","cbCaio7tB4Ry7KKE","https://ap.wps.com/l/cbCaio7tB4Ry7KKE","pdf",484758,10,"English","# 1 Vector and Matrix\n## Vectors\n## Matrix Examples and Representations\n## Linear Systems and Solution Types\n# 2 Matrix Operations\n# 3 Elimination\n## Augmented Matrix and Row Operations\n## Solving by Transforming [A|b] to [I|x]\n## Elimination and Permutation Matrices\n# Matrix Inverse","[{\"question\":\"How are vector dot product, length, and angle between two vectors computed in the cheat sheet?\",\"answer\":\"It gives the dot product as Ax·Bx + Ay·By + Az·Bz, length as ‖A‖ = √(Ax^2 + Ay^2 + Az^2), and the angle as θ = arccos((A·B)/(‖A‖‖B‖)).\"},{\"question\":\"What conditions determine whether the linear system Ax=b has no solution, a unique solution, or infinitely many solutions?\",\"answer\":\"It states Ax=b has no solution unless b lies in the span of the column vectors of A; in the unique-solution case, A must be square and its columns must be linearly independent, equivalently A has an inverse and x = A^{-1}b. Infinite solutions occur when the system is consistent but not uniquely determined (dependent columns / reduced row echelon form with free variables).\"},{\"question\":\"How does the cheat sheet describe Gaussian elimination using augmented matrices?\",\"answer\":\"It defines the augmented matrix [A|b], then uses row operations (scaling a row, replacing a row by a multiple plus another row, and swapping rows) to transform [A|b] toward [I|x]. It also explains elimination and permutation matrices that represent these row transformations.\"}]","Linear Algebra Cheat Sheet - January 2024 | PDF",1791926984]