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The text develops methods for solving systems of linear equations using Gaussian elimination, elementary row operations, and reduced row echelon form, followed by the simplex method for optimization via graphical and algorithmic approaches. It then treats vector spaces, linear transformations, determinants, subspaces, linear independence, bases and dimension, eigenvalues and eigenvectors, and diagonalization, emphasizing geometric and algebraic connections throughout.",{"@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/linear-algebra-a-course-on-vectors-matrices-and-linear-transformations/234491/",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-a-course-on-vectors-matrices-and-linear-transformations/234491.png","ImageObject",442,249,{"name":88,"@type":89},"Elsa","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-11","2026-09-11",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What topics are covered in the course introduction to linear algebra?","Question",{"text":108,"@type":109},"The introduction explains how to organize key concepts, introduces vectors and linear functions, and motivates matrices and matrix multiplication as a composition of functions.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the book approach solving systems of linear equations?",{"text":113,"@type":109},"It develops Gaussian elimination with augmented matrix notation, equivalence during solving, and reduced row echelon form, then extends the approach using elementary row operations and related solution-set geometry.",{"name":115,"@type":106,"acceptedAnswer":116},"Which optimization method is discussed and how is it developed?",{"text":117,"@type":109},"The simplex method is introduced through Pablo's problem, then explored using graphical solutions and Dantzig's algorithm, culminating in additional problem connections and review practice.","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},234491,1790292580,{"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":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"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":141},137455077381,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Linear Algebra  \nDavid Cherney, Tom Denton, Rohit Thomas and Andrew Waldron  \nEdited by Katrina Glaeser and Travis Scrimshaw First Edition. Davis California, 2013 .  \nThis work is licensed under a  \nCreative Commons Attribution-NonCommercialShareAlike 3 .0 Unported License.  \nContents  \n1 What is Linear Algebra? 9  \n1.1 Organizing Information ...................... 9  \n1.2 What are Vectors? ........................ 12  \n1.3 What are Linear Functions? ................... 15  \n1.4 So, What is a Matrix? ...................... 20  \n1.4.1 Matrix Multiplication is Composition of Functions ... 25  \n1.4.2 The Matrix Detour .................... 26  \n1.5 Review Problems ......................... 30  \n2 Systems of Linear Equations 37  \n2.1 Gaussian Elimination ....................... 37  \n2.1.1 Augmented Matrix Notation ............... 37  \n2.1.2 Equivalence and the Act of Solving ........... 40  \n2.1.3 Reduced Row Echelon Form ............... 40  \n2.1.4 Solution Sets and RREF ................. 45  \n2.2 Review Problems ......................... 48  \n2.3 Elementary Row Operations ................... 52  \n2.3.1 EROs and Matrices .................... 52  \n2.3.2 Recording EROs in (M jI ) ................ 54  \n2.3.3 The Three Elementary Matrices ............. 56  \n2.3.4 LU , LDU, and PLDU Factorizations .......... 58  \n2.4 Review Problems ......................... 61  \n2.5 Solution Sets for Systems of Linear Equations ......... 63  \n2.5.1 The Geometry of Solution Sets: Hyperplanes ...... 64  \n2.5.2 Particular Solution + Homogeneous Solutions ..... 65  \n2.5.3 Solutions and Linearity .................. 66  \n2.6 Review Problems ......................... 68  \n3 The Simplex Method 71  \n3.1 Pablo's Problem .......................... 71  \n3.2 Graphical Solutions ........................ 73  \n3.3 Dantzig's Algorithm ....................... 75  \n3.4 Pablo Meets Dantzig ....................... 78  \n3.5 Review Problems ......................... 80  \n4 Vectors in Space, n-Vectors 83  \n4.1 Addition and Scalar Multiplication in Rn ............ 84  \n4.2 Hyperplanes ............................ 85  \n4.3 Directions and Magnitudes .................... 88  \n4.4 Vectors, Lists and Functions: RS ................ 94  \n4.5 Review Problems ......................... 97  \n5 Vector Spaces 101  \n5.1 Examples of Vector Spaces .................... 102  \n5.1.1 Non-Examples ....................... 106  \n5.2 Other Fields ............................ 107  \n5.3 Review Problems ......................... 109  \n6 Linear Transformations 111  \n6.1 The Consequence of Linearity .................. 112  \n6.2 Linear Functions on Hyperplanes ................ 114  \n6.3 Linear Di􀀋erential Operators ................... 115  \n6.4 Bases (Take 1) .......................... 115  \n6.5 Review Problems ......................... 118  \n7 Matrices 121  \n7.1 Linear Transformations and Matrices .............. 121  \n7.1.1 Basis Notation ...................... 121  \n7.1.2 From Linear Operators to Matrices ........... 127  \n7.2 Review Problems ......................... 129  \n7.3 Properties of Matrices ...................... 133  \n7.3.1 Associativity and Non-Commutativity ......... 140  \n7.3.2 Block Matrices ...................... 142  \n7.3.3 The Algebra of Square Matrices ............ 143  \n7.3.4 Trace ............................ 145  \n7.4 Review Problems ......................... 146  \n7.5 Inverse Matrix ........................... 150  \n7.5.1 Three Properties of the Inverse ............. 150  \n7.5.2 Finding Inverses (Redux) ................. 151  \n7.5.3 Linear Systems and Inverses ............... 153  \n7.5.4 Homogeneous Systems .................. 154  \n7.5.5 Bit Matrices ........................ 154  \n7.6 Review Problems ......................... 155  \n7.7 LU Redux ............................. 159  \n7.7.1 Using LU Decomposition to Solve Linear Systems ... 160  \n7.7.2 Finding an LU Decomposition.............. 162  \n7.7.3 Block LDU Decomposition ................ 165  \n7.8 Review Problems ...............","cbCaipqqbwyakGjF","https://ap.wps.com/l/cbCaipqqbwyakGjF","pdf",5895711,436,"English","# What is Linear Algebra?\n## Organizing Information\n## What are Vectors?\n## What are Linear Functions?\n## So, What is a Matrix?\n## Review Problems\n# Systems of Linear Equations\n## Gaussian Elimination\n## Review Problems\n## Elementary Row Operations\n## Review Problems\n## Solution Sets for Systems of Linear Equations\n## Review Problems\n# The Simplex Method\n## Pablo's Problem\n## Graphical Solutions\n## Dantzig's Algorithm\n## Review Problems\n# Vector Spaces\n## Examples of Vector Spaces\n## Non-Examples\n## Other Fields\n## Review Problems\n# Linear Transformations\n## The Consequence of Linearity\n## Linear Functions on Hyperplanes\n## Linear Differential Operators\n## Bases (Take 1)\n## Review Problems\n# Matrices\n## Properties of Matrices\n## Inverse Matrix\n## LU Redux\n## Review Problems\n# Determinants\n## The Determinant Formula\n## Elementary Matrices and Determinants\n## Properties of the Determinant\n## Review Problems\n# Subspaces and Spanning Sets\n## Subspaces\n## Building Subspaces\n## Review Problems\n# Linear Independence\n## Showing Linear Dependence\n## Showing Linear Independence\n## Review Problems\n# Basis and Dimension\n## Bases in Rn\n## Matrix of a Linear Transformation (Redux)\n## Review Problems\n# Eigenvalues and Eigenvectors\n## Invariant Directions\n## The Eigenvalue-Eigenvector Equation\n## Eigenspaces\n## Review Problems\n# Diagonalization\n## Diagonalizability\n## Change of Basis\n## Review Problems\n# Orthonormal Bases and Complements","[{\"question\":\"What topics are covered in the course introduction to linear algebra?\",\"answer\":\"The introduction explains how to organize key concepts, introduces vectors and linear functions, and motivates matrices and matrix multiplication as a composition of functions.\"},{\"question\":\"How does the book approach solving systems of linear equations?\",\"answer\":\"It develops Gaussian elimination with augmented matrix notation, equivalence during solving, and reduced row echelon form, then extends the approach using elementary row operations and related solution-set geometry.\"},{\"question\":\"Which optimization method is discussed and how is it developed?\",\"answer\":\"The simplex method is introduced through Pablo's problem, then explored using graphical solutions and Dantzig's algorithm, culminating in additional problem connections and review practice.\"}]","Linear Algebra - A Course on Vectors, Matrices, and Linear Transformations | PDF",1789087838,153]