A First Course in LINEAR ALGEBRA - Lyryx Learning

1 With Open TextsA First Course inLINEAR ALGEBRAan Open TextBASE TEXTBOOKVERSION 2017 REVISION AADAPTABLE | ACCESSIBLE | AFFORDABLEby Lyryx Learningbased on the original text by K. KuttlerCreative Commons License (CC BY)advancing learningChampions of Access to KnowledgeOPEN TEXTONLINEASSESSMENTAll digital forms of access to our high-qualityopen texts are entirely FREE! All content isreviewed for excellence and is wholly adapt-able; custom editions are produced by Lyryxfor those adopting Lyryx assessment. Accessto the original source files is also open to any-one!

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4 This text, including the art and illustrations, are available underthe Creative Commons license (CC BY), allowing anyone to reuse, revise, remix and redistribute the view a copy of this license, learningA First Course in LINEAR Algebraan Open TextBase Text Revision HistoryCurrent Revision: Version 2017 Revision AExtensive edits, additions, and revisions have been completed by the editorial staff at Lyryx new content (text and images) is released under the same license as noted A Lyryx : Front matter has been updated including cover, copyright, and revision pages.

5 I. Farah: contributed edits and revisions, particularly the proofs in the Properties of Determinants II:Some Important Proofs section2016 B Lyryx : The text has been updated with the addition of subsections on Resistor Networks and theMatrix Exponential based on original material by K. Kuttler. Lyryx : New Random Walks A Lyryx : The layout and appearance of the text has been updated, including the title page and newlydesigned back A Lyryx : The content was modified and adapted with the addition of new material and several im-agesthroughout.

6 Lyryx : Additional examples and proofs were added to existing material A Original textby K. Kuttler of Brigham Young University. That version is used under Creative Com-mons license CC BY ( ) made possible byfunding from The Saylor Foundation s Open Textbook Challenge. SeeElementary LINEAR Algebraformore information and the original Systems of Systems of Equations, Geometry.. Systems Of Equations, Algebraic Procedures.. Elementary Operations.. Gaussian Elimination.. Uniqueness of the Reduced Row-Echelon Form.. Rank and Homogeneous Systems.

7 Balancing Chemical Reactions.. Dimensionless Variables.. An Application to Resistor Networks.. 382 Matrix Arithmetic.. Addition of Matrices.. Scalar Multiplication of Matrices.. Multiplication of Matrices.. Thei jthEntry of a Product.. Properties of Matrix Multiplication.. The Transpose.. The Identity and Inverses.. Finding the Inverse of a Matrix.. Elementary Matrices.. More on Matrix Inverses.. Finding AnLUFactorization By Inspection.. , Multiplier Method.. Solving Systems usingLUFactorization.

8 Justification for the Multiplier Method.. 102iiiivCONTENTS3 Basic Techniques and Properties.. Cofactors and 2 2 Determinants.. The Determinant of a Triangular Matrix.. Properties of Determinants I: Examples.. Properties of Determinants II: Some Important Proofs.. Finding Determinants using Row Operations.. Applications of the Determinant.. A Formula for the Inverse.. Cramer s Rule.. Polynomial Interpolation.. Vectors inRn.. ALGEBRA inRn.. Addition of Vectors inRn.. Scalar Multiplication of Vectors inRn.

9 Geometric Meaning of Vector Addition.. Length of a Vector.. Geometric Meaning of Scalar Multiplication.. Parametric Lines.. The Dot Product.. The Dot Product.. The Geometric Significance of the Dot Product.. Projections.. Planes inRn.. The Cross Product.. The Box Product.. Spanning, LINEAR Independence and Basis inRn.. Spanning Set of Vectors.. Linearly Independent Set of Vectors.. A Short Application to Chemistry.. Subspaces and Basis.. Row Space, Column Space, and Null Space of a Matrix.. Orthogonality and the Gram Schmidt Process.

10 Orthogonal and Orthonormal Sets.. Orthogonal Matrices.. Gram-Schmidt Process.. Orthogonal Projections.. Least Squares Approximation.. Applications.. Vectors and Physics.. Work.. 2645 LINEAR LINEAR Transformations.. The Matrix of a LINEAR Transformation I.. Properties of LINEAR Transformations.. Special LINEAR Transformations inR2.. One to One and Onto Transformations.. Isomorphisms.. The Kernel And Image Of A LINEAR Map.. The Matrix of a LINEAR Transformation II.. The General Solution of a LINEAR System.