[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572017-105":53,"doc-detail-572017-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","ma2101-linear-algebra-ii-fields-vector-spaces","MA2101 Linear Algebra II - Fields & Vector Spaces","","MA2101 Linear Algebra II focuses on foundational algebraic structures and their vector-space consequences. It defines fields, rings, and groups in terms of closure, identity, associativity, invertibility, commutativity, and distributivity, then introduces vector spaces via the vector-space axioms. The notes develop key concepts of subspaces, including equivalent definitions and the requirement that subspaces contain the zero vector. They also cover spans, sums and intersections of subspaces, direct sums, linear independence, bases, and dimension, followed by matrix operations and determinant criteria for invertibility.",{"@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/ma2101-linear-algebra-ii-fields-vector-spaces/572017/",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/ma2101-linear-algebra-ii-fields-vector-spaces/572017.png","ImageObject",442,249,{"name":88,"@type":89},"Valentina","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-23","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What axioms define a vector space in these notes?","Question",{"text":109,"@type":110},"A vector space satisfies closure, identity, associativity, invertibility, distributivity, and commutativity for vector addition and scalar multiplication. The notes summarize these requirements as the vector-space axioms.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Why must a subspace contain the zero vector?",{"text":114,"@type":110},"The notes state that any subspace must include 0W, equivalently 0V. This is required for the subspace to be closed under the relevant operations used in the definitions.",{"name":116,"@type":107,"acceptedAnswer":117},"How do determinants relate to invertibility of a square matrix?",{"text":118,"@type":110},"The notes give the criterion that a square matrix is invertible if and only if its determinant is nonzero. They also list equivalences such as the existence of an inverse via AB = I.","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},572017,1792036234,{"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":73,"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":9},13056703020460,"https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923","MA2101 Linear Algebra II Fields & Vector Spaces  \n❼ Def 1.3 [Field, ring, group] :  \nClosure, identity, associativity, invertibility, commutativity for + and 􀀂 (invertibility for 􀀂 excludes 0), distributivity of 􀀂 over +  \nFields satisfy all axioms; (commutative) rings satisfy all except 􀀂 invertibility; (additive) groups do not have 􀀂 and may not satisfy + commutativity  \n❼ Def 1.12 [Vector space]: Satis􀀌es closure, identity,  \nassociativity, invertibility, distributivity, commutativity ❼ Ex 1.18 [Matrix space] : Mn (F ), Mm􀀂n(F ) | m rows  \nVector Subspaces  \n❼ Def 2.1 [Subspace] : Set closed under vec. + and scalar 􀀂❼ Rem 2.3: Subspace must have 0W (= 0V )  \n❼ Thm 2.8 [Equiv. subspace def.] : W is subspace of V () W is closed under linear combination () W with + and 􀀂 of V becomes a vector space  \n❼ Ex 2.9: Null space of matrix A is a subspace, [i.e. solutions](i.e. solutions)[ ](i.e. solutions)to homogeneous linear equations form a subspace  \n❼ Thm 2.15: The intersection of (possibly in􀀌nite number of) subspaces is a subspace: T􀀋2I W􀀋 is a subspace  \n❼ LA1 Ex [3.1.8.3](3.1.8.3) [2D & 3D geometry] : Line in R2  : f (x; y) j ax + by = cg  \nPlane in R3  : f (x; y; z) j ax + by + cz = dg alternatively: (1) 􀀈􀀀 ~~d~~􀀀~~bs~~a􀀀~~ct~~ ; s; t 􀀁 j s; t 2 R􀀉 if a  0  \n(2) 􀀈􀀀s; ~~d~~􀀀~~as~~b􀀀~~ct~~ ; t 􀀁 j s; t 2 R􀀉 if b  0  \n(3) 􀀈􀀀s; t; ~~d~~􀀀~~as~~c􀀀~~bt~~ 􀀁 j s; t 2 R􀀉 if c  0 Line in R3  : f (a0 ; b0 ; c0 ) + t(a; b; c) j t 2 Rg  \nSums of Subspaces  \n❼ Ex 3.3: S 􀀒 T =) Span(S) 􀀒 Span(T )  \n❼ Thm 3.4: S is subset of V , then Span(S) is subspace of V , and it is the smallest subspace containing S  \n❼ Tut 2.5: For subsets Ui ; Wi ; U, and subspace W:  \n(1) U1 􀀒 U2 and W1 􀀒 W2 =) U1 + W1 􀀒 U2 + W2  \n(2) W + f0g = W , W + W = W  \n(3) U + W = W () U 􀀒 W  \n❼ Thm 3.11: For subspaces U; W 􀀒 V:  \n(1) U + W = Span(U [ W )  \n(2) U + W is a subspace of V  \n(3) U + W is the smallest subspace containing both U , W ❼ Thm 3.18: (induction of Thm 3.11) For subspaces Ui 􀀒 V:  \n(1) Pi Wi = Span( Si Wi)  \n(2) Pi Wi is a subspace of V  \n(3) Pi Wi is the smallest subspace containing all Wi  \n❼ Ex 3.15: For subspaces U; W 􀀒 V:  \nU [ W is subspace () U 􀀒 W or W 􀀒 U ❼ Ex 3.16: Span(S [ T ) = Span(S) + Span(T )  \n❼ Def 3.20 [Direct sum of subspaces] :  \nW = W1 􀀈 W2 () W = W1 + W2 and W 1 \\ W2 = f0g  \n❼ Thm 3.25 [[Equiv. direct](Equiv. direct) sum def.]: Wi are subspacesof V , W = W1 + W2 :  \nW1 + W2 is a direct sum () 8w 2 W; w = w 1 + w2  \nwhere w 1 2 W1 , w2 2 W2 and the w 1 , w2 are unique ❼ Def 3.26 [Direct sum of many subspaces] :  \nW = MWi () W = XWi and  \ni i  \n8k; kXi11 Wi! \\ Wk = f0g  \n❼ Thm 3.31 [Equiv. direct sum de􀀌nition] :  \ns  \nIf Wi are subspaces of vector space V , W = XWi then:  \ns i=1  \nX Wi is a direct sum of Wi  \ni=1  \n() 8l; Xil Wi  \\ Wl = f0g  \ns  \n() all wi are unique in w = X wi (where wi 2 Wi)  \ni=1  \nLinear Independence  \n❼ Def 4.1 [Linear independence] : S = fvig is a linearly independent set () (Pi xivi = 0 =) all xi = 0)  \n❼ Thm 4.8 [Equiv. linear independence def.] :  \n(1) S = fvig is linearly independent  \n() no vk 2 S is a linear combination of others  \n(2) S = fv1 g is linearly independent () v 1  0  \n(3) S = fv1 ; v2 g is linearly independent  \n() neither of v 1 , v2 is a scalar multiple of the other  \nBasis & Dimension  \n❼ Def 4.10 [Basis] : B 􀀒 V is a basis  \n() V = Span(B) and B is a linearly independent set =) dimF V := j Bj (for 􀀌nite cardinality of B)  \n❼ Thm 4.11 [Equiv. basis def. I] : B = vi is a basis of V  \n() 8v 2 V , v = Pi ai vi uniquely () V = Li Spanfvig  \n❼ Thm 4.13 [Basis from spanning set] : V = Span(B)=) 9B1 􀀒 B such that B 1 is a basis of V , and  \nany maximal linear independent set B2 􀀒 B is a basis of V ❼ Thm 4.15 [Dimension is well-de􀀌ned] :  \nIf B is a basis of V then:  \n(1) jSj > j Bj =) S is linearly dependent  \n(2) jTj \u003C jBj =) T does not span V  \n(3) B 0 is another basis of V =) jB0 j = j Bj (= dimF V )  \n❼ Thm 4.16 [Expanding a linearly independent set] :  \nIf B 􀀒 V is linearly independent then","cbCaigBQOip7pERD","https://ap.wps.com/l/cbCaigBQOip7pERD","pdf",341660,"English","# Vector spaces and subspaces\n## Fields and vector space axioms\n## Subspaces, null space, and equivalent definitions\n## Sums, intersections, and direct sums\n## Linear independence, basis, and dimension\n# Matrix manipulation and determinants\n## Transposition and inverse properties\n## Cofactor expansion\n## Triangular determinants and invertibility","[{\"question\":\"What axioms define a vector space in these notes?\",\"answer\":\"A vector space satisfies closure, identity, associativity, invertibility, distributivity, and commutativity for vector addition and scalar multiplication. The notes summarize these requirements as the vector-space axioms.\"},{\"question\":\"Why must a subspace contain the zero vector?\",\"answer\":\"The notes state that any subspace must include 0W, equivalently 0V. This is required for the subspace to be closed under the relevant operations used in the definitions.\"},{\"question\":\"How do determinants relate to invertibility of a square matrix?\",\"answer\":\"The notes give the criterion that a square matrix is invertible if and only if its determinant is nonzero. They also list equivalences such as the existence of an inverse via AB = I.\"}]","MA2101 Linear Algebra II - Fields & Vector Spaces | PDF",1791925065]