Introduction to Linear Algebra

Linear Algebra with Applications is an introductory text targeted to secondor advanced first year undergraduate students. The organization of this textis motivated by what our experience tells us are the essential concepts thatstudents should master in a one semester undergraduate Linear Algebra course.The centerpiece of our philosophy regarding the presentation of the materialis that each topic should be fully developed before moving on to the next. Inaddition, there should be a natural connection between topics. We take greatcare to meet both of these objectives. This allows us to stay on task so thateach topic can be covered with the depth required before progressing to the nextlogical one. As a result the reader is prepared for each new unit and there is noneed to repeat a concept in a subsequent chapter when it is utilized. Linear Algebra is taken early in an undergraduate curriculum and yet offersthe opportunity to introduce the importance of abstraction, not only in mathematics, but in many other areas where Linear Algebra is used. Our approachis to take advantage of this opportunity by presenting abstract vector spacesas early as possible. Throughout the text, we are mindful of the difficultiesthat students at this level have with abstraction and introduce new conceptsfirst through examples which gently illustrate the idea. To motivate the defini-tion of an abstract vector space, and the subtle concept of linear independence,we use addition and scalar multiplication of vectors in Euclidean Space. Wehave strived to create a balance between computation, problem solving, and ab-straction. This approach equips students with the necessary skills and problemsolving strategies in an abstract setting that allows for a greater understandingand appreciation for the numerous applications of the subject.

Chapter 1 Systems of Linear Equations and Matrices 1 - 1.1 Systems of Linear Equations Exercise Set 1.1 - 1.2 Matrices and Elementary Row Operations Exercise Set 1.2 - 1.3 Matrix Algebra Exercise Set 1.3 - 1.4 The Inverse of a Square Matrix Exercise Set 1.4 - 1.5 Matrix Equations Exercise Set 1.5 - 1.6 Determinants Exercise Set 1.6 - 1.7 Elementary Matrices and LU Factorization Exercise Set 1.7 - 1.8 Applications of Systems of Linear Equatio Exercise Set 1.8 Review Exercises Chapter Test Chapter 2 Linear Combinations and Linear Independence - 2.1 Vectors in Rn Exercise Set 2.1 - 2.2 Linear Combinations Exercise Set 2.2 - 2.3 Linear Independence Exercise Set 2.3 Review Exercises Chapter Test Chapter 3 Vector Spaces - 3.1 Definition of a Vector Space Exercise Set 3.1 - 3.2 Subspaces Exercise Set 3.2 - 3.3 Basis and Dimension Exercise Set 3.3 - 3.4 Coordinates and Change of Basis Exercise Set 3.4 - 3.5 Application : Differential Equations Exercise Set 3.5 Review Exercises Chapter Test Chapter 4 Linear Transformations - 4.1 Linear Transformations Exercise Set 4.1 - 4.2 The Null Space and Range Exercise Set 4.2 - 4.3 Isomorphisms Exercise Set 4.3 - 4.4 Matrix Representation of a Linear Transformation Exercise Set 4.4 - 4.5 Similarity Exercise Set 4.5 - 4.6 Application : Computer Graphics Exercise Set 4.6 Review Exercises Chapter Test Chapter 5 Eigenvalues and Eigenvectors - 5.1 Eigenvalues and Eigenvectors Exercise Set 5.1 - 5.2 Diagonalization Exercise Set 5.2 - 5.3 Application : Systems of Linear Different Exercise Set 5.3 - 5.4 Application : Markov Chains Exercise Set 5.4 Review Exercises Chapter Test Chapter 6 Inner Product Spaces - 6.1 The Dot Product on Rn Exercise Set 6.1 - 6.2 Inner Product Spaces Exercise Set 6.2 - 6.3 Orthonormal Bases Exercise Set 6.3 - 6.4 Orthogonal Complements Exercise Set 6.4 - 6.5 Application : Least Squares Approximation Exercise Set 6.5 - 6.6 Diagonalization of Symmetric Matrices Exercise Set 6.6 - 6.7 Application : Quadratic Forms Exercise Set 6.7 - 6.8 Application : Singular Value Decomposition Exercise Set 6.8 Review Exercises Chapter Test A Preliminaries A.1 Algebra of Sets Exercise Set A.1 A.2 Functions Exercise Set A.2 A.3 Techniques of Proof Exercise Set A.3 A.4 Mathematical Induction Exercise Set A.4 Answers to Odd-Numbered Exercises A.3 Techniques of Proof Exercise Set A.3 A.4 Mathematical Induction Exercise Set A.4 Answers to Odd-Numbered Exercises