Introduction to Linear Algebra
Published on 16. August 2009
Book
416 pages
978-0-07-712715-2 (ISBN)
Description
Linear Algebra with Applications is an introductory text targeted to second or advanced first year undergraduates in engineering or mathematics. The organization of this text is motivated by the authors' experience which tells them what essential concepts should be mastered by students in a one semester undergraduate Linear Algebra course. The authors' main objectives are to fully develop each topic before moving on and to connect topics naturally. The authors take great care to meet both these objectives, because this organization will allow instructors teaching from this text to stay on task so that each topic can be covered with the depth required before progressing to the next logical one. As a result the reader is prepared for each new unit and there is no need to repeat a concept in a subsequent chapter when it is utilized.
This text is geared towards an introductory linear algebra course taken by first or second year undergraduate students. However, it offers the opportunity to introduce the importance of abstraction, not only in mathematics, but in many other areas where Linear Algebra is used. The textbook's approach is to take advantage of this opportunity by presenting abstract vector spaces as early as possible. Throughout the text, the authors are mindful of the difficulties that students at this level have with abstraction and introduce new concepts first through examples which gently illustrate the idea. To motivate the definition of an abstract vector space, and the subtle concept of linear independence, the authors use addition and scalar multiplication of vectors in Euclidean Space. The authors have strived to create a balance between computation, problem solving, and abstraction. This approach equips students with the necessary skills and problem solving strategies in an abstract setting that allows for a greater understanding and appreciation for the numerous applications of the subject.
This text is geared towards an introductory linear algebra course taken by first or second year undergraduate students. However, it offers the opportunity to introduce the importance of abstraction, not only in mathematics, but in many other areas where Linear Algebra is used. The textbook's approach is to take advantage of this opportunity by presenting abstract vector spaces as early as possible. Throughout the text, the authors are mindful of the difficulties that students at this level have with abstraction and introduce new concepts first through examples which gently illustrate the idea. To motivate the definition of an abstract vector space, and the subtle concept of linear independence, the authors use addition and scalar multiplication of vectors in Euclidean Space. The authors have strived to create a balance between computation, problem solving, and abstraction. This approach equips students with the necessary skills and problem solving strategies in an abstract setting that allows for a greater understanding and appreciation for the numerous applications of the subject.
More details
Content
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
A complete book that students can read independently from start to finish.
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
A complete book that students can read independently from start to finish.