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!We have been developing superior online for-mative assessment for more than 15 years. Ourquestions are continuously adapted with thecontent and reviewed for quality and soundpedagogy. To enhance Learning , students re-ceive immediate personalized feedback. Stu-dent grade reports and performance statisticsare also to our in-house support team is avail-able 7 days/week to provide prompt resolutionto both student and instructor inquiries.

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3 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. 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. Lyryx : Additional examples and proofs were added to existing material A Original textby K.

4 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.. 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.

5 Elementary Matrices.. More on Matrix Inverses.. Finding AnLUFactorization By Inspection.. , Multiplier Method.. Solving Systems usingLUFactorization.. 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.. Geometric Meaning of Vector Addition.. Length of a Vector.. Geometric Meaning of Scalar Multiplication.. Parametric Lines.

6 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.. 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.

7 Isomorphisms.. The Kernel And Image Of A LINEAR Map.. The Matrix of a LINEAR Transformation II.. The General Solution of a LINEAR System.. 3216 Complex Complex Numbers.. Polar Form.. Roots of Complex Numbers.. The Quadratic Formula.. 3437 Spectral Eigenvalues and Eigenvectors of a Matrix.. Definition of Eigenvectors and Eigenvalues.. Finding Eigenvectors and Eigenvalues.. Eigenvalues and Eigenvectors for Special Types of Matrices.. Diagonalization.. Similarity and Diagonalization.. Diagonalizing a Matrix.. Complex Eigenvalues.. Applications of Spectral Theory.. Raising a Matrix to a High Power.. Raising a Symmetric Matrix to a High Power.. Markov Matrices.. Eigenvalues of Markov Matrices.. Dynamical Systems.. The Matrix Exponential.. Orthogonality.. Orthogonal Diagonalization.

8 The Singular Value Decomposition.. Positive Definite Matrices.. The Cholesky Factorization.. TheQRFactorization and Eigenvalues.. Power Methods.. Quadratic Forms.. 4278 Some Curvilinear Coordinate Polar Coordinates and Polar Graphs.. Spherical and Cylindrical Coordinates.. 4499 Vector Algebraic Considerations.. Spanning Sets.. LINEAR Independence.. Subspaces and Basis.. Sums and Intersections.. LINEAR Transformations.. Isomorphisms.. One to One and Onto Transformations.. Isomorphisms.. The Kernel And Image Of A LINEAR Map.. The Matrix of a LINEAR Transformation.. 524A Some Prerequisite Sets and Set Notation.. Well Ordering and Induction.. 539B Selected Exercise Answers543 Index591 PrefaceA First Course in LINEAR Algebrapresents an introduction to the fascinating subject of LINEAR ALGEBRA forstudents who have a reasonable understanding of basic ALGEBRA .

9 Major topics of LINEAR ALGEBRA are pre-sented in detail, with proofs of important theorems provided. Separate sections may be included in whichproofs are examined in further depth and in general these canbe excluded without loss of possible, applications of key concepts are an effort to assist those students who areinterested in continuing on in LINEAR ALGEBRA connections to additional topics covered in advanced coursesare chapter begins with a list of desired outcomes which a student should be able to achieve uponcompleting the chapter. Throughout the text, examples and diagrams are given to reinforce ideas andprovide guidance on how to approach various problems. Students are encouraged to work through thesuggested exercises provided at the end of each section. Selected solutions to these exercises are given atthe end of the this is an open text, you are encouraged to interact with the textbook through annotating, revising,and reusing to your Systems of Systems of Equations, GeometryOutcomesA.

10 Relate the types of solution sets of a system of two (three)variables to the intersections oflines in a plane (the intersection of planes in three space)As you may remember, LINEAR equations like 2x+3y=6 can be graphed as straight lines in the coordi-nate plane. We say that this equation is in two variables, in this casexandy. Suppose you have two suchequations, each of which can be graphed as a straight line, and consider the resulting graph of two would it mean if there exists a point of intersection between the two lines? This point, which lies onbothgraphs, givesxandyvalues for which both equations are true. In other words, this point gives theordered pair (x,y) that satisfy both equations. If the point(x,y)is a point of intersection, we say that(x,y)is asolutionto the two equations. In LINEAR ALGEBRA , we often are concerned with finding the solution(s)to a system of equations, if such solutions exist.