[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572368-105":53,"doc-detail-572368-en":126},{"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":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","reference-sheet-chapter-1-section-11-13","REFERENCE SHEET - CHAPTER 1 - SECTION 1.1 - 1.3","","A mathematics reference sheet compiling key definitions and theorems for linear algebra topics across multiple chapters and sections. It organizes essential concepts such as ordered tuples in Rn, solutions to linear systems, elementary row operations, row echelon and reduced row echelon forms, pivot and free variables, and parametric solution structure. It also summarizes foundational facts about consistency, uniqueness, and infinite solutions, along with vector and matrix interpretations used throughout the course.",{"@graph":63,"@context":118},[64,80,101],{"@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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/reference-sheet-chapter-1-section-11-13/572368/",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/reference-sheet-chapter-1-section-11-13/572368.png","ImageObject",442,249,{"name":88,"@type":89},"Aditya","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What do the elementary row operations include?","Question",{"text":108,"@type":109},"They include scaling a row by a nonzero number, replacing one row with itself plus a multiple of another row, and swapping two rows. These operations are used to transform matrices in row reduction.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How are consistency and solution types determined for a linear system?",{"text":113,"@type":109},"A system is inconsistent if it has no solutions; it is consistent otherwise. For solution sets, the reference sheet distinguishes between zero solutions (inconsistent), a unique solution (all but the last column are pivot columns), and infinitely many solutions (some columns besides the last are not pivot columns, corresponding to free variables).",{"name":115,"@type":106,"acceptedAnswer":116},"What is the meaning of row echelon form, reduced row echelon form, pivot, and free variable?",{"text":117,"@type":109},"Row echelon form requires zero rows at the bottom, leading nonzero entries moving rightward down the rows, and zeros below each leading entry. Reduced row echelon form adds that each pivot equals 1 and is the only nonzero entry in its column. A pivot is the first nonzero entry in a row in row echelon form, and a free variable corresponds to a non-pivot column in the reduced row echelon form.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572368,1792068188,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":35,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":140},962085564549,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","IMPORTANT DEFINITIONS AND THEOREMS  \nREFERENCE SHEET  \nThis is a (not quite comprehensive) list of definitions and theorems given in Math 1553 . Pay particular attention to the ones in red.  \n\n|  | Study Tip |  |\n| --- | --- | --- |\n| For each definition, find an example of something that satisfies the requirements of the definition, and an example of something that does not. For each theorem, find an example of something that satisfies the hypotheses of the theorem, and an example of something that does not satisfy the conclusions (or the hypotheses, of course) of the theorem. This is great conceptual practice. |  |  |\n\nCHAPTER 1  \nSECTION 1.1.  \nDefinition. Rn = all ordered n-tuples of real numbers (x 1 , x2 , x3 , . . . , xn ) .  \nThis is the number line when n = 1, the xy-plane when n = 2, and space when n = 3. Definition. A solution to a system of linear equations is a list of numbers making all of the equations true.  \nA solution of a system of equations in n variables is a point in Rn .  \nDefinition. The elementary row operations are the following matrix operations:  \n􀀏 Multiply all entries in a row by a nonzero number (scale) .  \n􀀏 Add (a multiple of) each entry of one row to the corresponding entry in another (row replacement) .  \n􀀏 Swap two rows.  \nDefinition. Two matrices are called row equivalent if one can be obtained from the other by doing some number of elementary row operations.  \nDefinition. A system of equations is called inconsistent if it has no solution. It is consistent otherwise.  \nSECTION 1.2.  \nDefinition. A matrix is in row echelon form if (1) All zero rows are at the bottom.  \n2 DEFINITIONS AND THEOREMS  \n(2) Each leading nonzero entry of a row is to the right of the leading entry of the row above.  \n(3) Below a leading entry of a row, all entries are zero.  \nDefinition. A pivot is the first nonzero entry of a row of a matrix in row echelon form.  \nDefinition. A matrix is in reduced row echelon form if it is in row echelon form, and in addition,  \n(4) The pivot in each nonzero row is equal to 1 .  \n(5) Each pivot is the only nonzero entry in its column.  \nTheorem. Every matrix is row equivalent to one and only one matrix in reduced row echelon form.  \nReview. Row reduction algorithm.  \nSECTION 1.3.  \nDefinition. Consider a consistent linear system of equations in the variables x 1 , . . . , xn . Let A be the reduced row echelon form of the matrix for this system. We say that xi is a free variable if its corresponding column inA is not a pivot column.  \nDefinition. The parametric form for the general solution to a system of equations is a system of equations for the non-free variables in terms of the free variables. For instance, if x2 and x4 are free,  \nx 1 = 2 􀀀 3x4 x3 = 􀀀1 􀀀 4x4  \nis a parametric form.  \nTheorem. Every solution to a consistent linear system is obtained by substituting (unique) values for the free variables in the parametric form.  \nFact. There are three possibilities for the solution set of a linear system with augmented matrix A:  \n(1) The system is inconsistent: it has zero solutions, and the last column of A is a pivot column.  \n(2) The system has a unique solution: every column of A except the last is a pivot column.  \n(3) The system has infinitely many solutions: the last column isn't a pivot column, and some other column isn't either. These last columns correspond to free variables.  \nCHAPTER 2  \nSECTION 2.1.  \nDefinition. A vector is an arrow with a given length and direction.  \nDefinition. A scalar is another name for a real number (to distinguish it from a vector) .  \nReview. Parallelogram law for vector addition, geometric interpretation of vector subtraction and scalar multiplication.  \nDEFINITIONS AND THEOREMS 3  \nDefinition. A linear combination of vectors v1 , v2 , . . . , vn is a vector of the form  \nc1 v1 + c2 v2 + 􀀁 􀀁 􀀁 + cn vn  \nwhere c1 , c2 , . . . , cn are scalars, called the weights or coefficients of the linear combination.  \nSECTION 2.2.  \nDefinition. A vector equation is an equat","cbCaia73EKm6a8yS","https://ap.wps.com/l/cbCaia73EKm6a8yS","pdf",127352,"English","# CHAPTER 1\n## SECTION 1.1\n## SECTION 1.2\n## SECTION 1.3\n# CHAPTER 2\n## SECTION 2.1\n## SECTION 2.2\n## SECTION 2.3\n## SECTION 2.4","[{\"question\":\"What do the elementary row operations include?\",\"answer\":\"They include scaling a row by a nonzero number, replacing one row with itself plus a multiple of another row, and swapping two rows. These operations are used to transform matrices in row reduction.\"},{\"question\":\"How are consistency and solution types determined for a linear system?\",\"answer\":\"A system is inconsistent if it has no solutions; it is consistent otherwise. For solution sets, the reference sheet distinguishes between zero solutions (inconsistent), a unique solution (all but the last column are pivot columns), and infinitely many solutions (some columns besides the last are not pivot columns, corresponding to free variables).\"},{\"question\":\"What is the meaning of row echelon form, reduced row echelon form, pivot, and free variable?\",\"answer\":\"Row echelon form requires zero rows at the bottom, leading nonzero entries moving rightward down the rows, and zeros below each leading entry. Reduced row echelon form adds that each pivot equals 1 and is the only nonzero entry in its column. A pivot is the first nonzero entry in a row in row echelon form, and a free variable corresponds to a non-pivot column in the reduced row echelon form.\"}]","REFERENCE SHEET - CHAPTER 1 - SECTION 1.1 - 1.3 | PDF",1791926988,6]