[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572370-105":53,"doc-detail-572370-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","basic-linear-algebra-review","Basic Linear Algebra Review","","Math 150-01 Linear Algebra final exam review sheet covering core concepts in vector and matrix operations. Topics include vector addition, scalar multiplication, linear combinations, dot products, vector length, unit vectors, angles, and orthogonality. Matrix sections cover matrix addition, scalar multiplication, matrix multiplication criteria and entries, non-commutativity, transposes, and reduced row echelon form (RREF) rules. It also summarizes special matrices, elementary matrices, and conditions for inverses.",{"@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/basic-linear-algebra-review/572370/",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/basic-linear-algebra-review/572370.png","ImageObject",442,249,{"name":88,"@type":89},"Nguyễn Văn Học","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",9,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How do you compute the dot product of two vectors?","Question",{"text":109,"@type":110},"The dot product is defined only when vectors have the same number of components. It equals the sum of the products of corresponding components.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What conditions make a matrix be in reduced row echelon form (RREF)?",{"text":114,"@type":110},"A matrix is in RREF when all nonzero rows are above zero rows, each pivot is to the right of pivots above it, all entries above and below each pivot are 0, and pivot entries are all 1.",{"name":116,"@type":107,"acceptedAnswer":117},"When is matrix multiplication AB defined, and how is an entry computed?",{"text":118,"@type":110},"AB is defined when the number of columns of A matches the number of rows of B. The (i,j) entry of AB is the dot product of the ith row of A and the jth column of B.","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},572370,1792013033,{"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":8,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},1374402739827,"https://ap-avatar.wpscdn.com/avatar/14000c97e7351f1a627?x-image-process=image/resize,m_fixed,w_180,h_180&k=1787885694763230660","Math 150-01 Linear Algebra  \nFinal Exam Review Sheet  \nVector Operations  \n• Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of components. Their sum is another vector, whose ith component is the sum of the ith components of v and w:  \n􀀔12􀀕 + 􀀔34􀀕 = 􀀔12  34􀀕  \n• Vectors may be multiplied by a scalar. The product cv is a vector, whose ith component is the product of c and the ith component of v:  \n5 􀀔12􀀕 = 􀀔 ((55)) ((12􀀕  \n• A linear combination of vectors is a sum of scalar multiples of vectors (with the same number of components):  \n5 􀀔12􀀕 + 6 􀀔40􀀕􀀀 3 􀀔 1􀀕 + 12 􀀔 3􀀕  \n• The dot product of two vectors is a scalar, and is only de􀀌ned when the vectors have the same number of components. It is the sum of the products of the ith components of the vectors:  \n􀀔12􀀕􀀁 􀀔34􀀕 = (1)(3) + (2)(4)  \n• The length of a vector is a scalar. It is found by taking the square root of the sum of the squares of the vector's components:  \n􀀍 2123 􀀍 = p 12 + 22 + 32 􀀍 435 􀀍  \n• A unit vector is a vector of length one. For example,[1 ; 0 ; 0] and [1= p2 ; 1= p 2] .  \n• The angle 􀀒 between two vectors v and w is related to their lengths and dot products by the formula v 􀀁 w  \ncos(􀀒) = jj v jj jj w jj  \n• Two vectors are perpendicular (also called orthogonal) if they have a dot product of 0 .  \nMatrix Operations  \n• Matrix addition is a component-wise operation. Two matrices A and B may be added together as long as they have the same number of rows and columns. Their sum is another matrix, whose (i; j)th entry is the sum of the (i; j)th entries of A and B:  \n􀀔13 24􀀕 + 􀀔57 68􀀕 = 􀀔13  57 24  68􀀕  \n• Matrices may be multiplied by a scalar. The product cA is a matrix, whose (i; j)th entry is the product of c and the (i; j)th entry of A:  \n5 􀀔13 24􀀕 = 􀀔 ((55)) ((13)) ((55)) ((24􀀕  \n• Matrices may be multiplied as long as their dimensions are compatible. The product AB is de􀀌ned when A has n columns and B has n rows. The (i; j)th entry of AB is the dot product of the ithrow of A and the jth column of B:  \n􀀔14  \n2 5  \n36􀀕  \n2 79 411  \n8101235 = 􀀔 ((14)) ((77))  ((25)) ((99))  ((36)) (( 1111)) ((14)) ((88))  ((25)) (( 1010))  ((36)) (( 1212􀀕  \n• Matrix multiplication is not commutative. In general, AB  BA.  \n• The transpose of a matrix A, denoted AT, is the matrix whose (i; j)th entry is the (j; i)th entry of A:  \nA = 􀀔14 25 36􀀕 ) AT = 24123 45635  \nReduced Echelon Form  \n• A matrix is in reduced row echelon form (RREF) when  \n1. All rows containing nonzero entries are above any rows containing only zero  \n2. The 􀀌rst nonzero entry (from the left) in each row, called a pivot, is strictly to the right of the 􀀌rst nonzero entry of any rows above it  \n3. All entries above and below a pivot are 0  \n4. The pivot entries are all 1  \nThe following matrices are in reduced row echelon form:  \n210 01 00 233 210 01 00 13 210 01 32 23 210 00 01 833 40 0 1 55 ; 40 0 0 0 5 ; 40 0 0 0 5 ; 40 0 0 15  \n• A matrix can be brought to its RREF by elementary row operations:  \n1. interchanging two rows  \n2. adding a multiple of one row to another  \n3. multiplying a row by a scalar  \n• The rank of a matrix is the number of pivots that appear in its RREF. The matrices above have rank 3, 2, 2, and 3, respectively.  \nSome Special Matrices  \n• The identity matrix In is an n 􀀂 n matrix whose diagonal entries are 1, and whose remaining entries are 0 . It is the multiplicative identity, because if A is any m 􀀂 n matrix, AIn = ImA = A. When the size n is understood, we can simply write I for the identity matrix.  \n• A zero matrix 0 is a matrix with every entry 0 . It can be any size, and not necessarily square. A zero matrix is never invertible. It is the additive identity, since A + 0 = 0 + A = A for any matrix A.  \n• An elementary matrix E is a square matrix which performs a row operation on a matrix A. There are three types of elementary matrices, all of which are invertible:  \n1. E can interchange two rows. An ","cbCaihhIlaklqkGw","https://ap.wps.com/l/cbCaihhIlaklqkGw","pdf",149975,"English","# Vector Operations\n## Matrix Operations\n# Reduced Echelon Form\n# Some Special Matrices\n# Matrix Inverses","[{\"question\":\"How do you compute the dot product of two vectors?\",\"answer\":\"The dot product is defined only when vectors have the same number of components. It equals the sum of the products of corresponding components.\"},{\"question\":\"What conditions make a matrix be in reduced row echelon form (RREF)?\",\"answer\":\"A matrix is in RREF when all nonzero rows are above zero rows, each pivot is to the right of pivots above it, all entries above and below each pivot are 0, and pivot entries are all 1.\"},{\"question\":\"When is matrix multiplication AB defined, and how is an entry computed?\",\"answer\":\"AB is defined when the number of columns of A matches the number of rows of B. The (i,j) entry of AB is the dot product of the ith row of A and the jth column of B.\"}]","Basic Linear Algebra Review | PDF",1791926993]