An Introduction to Linear Algebra
Published on 8. December 2022
236 pages
978-2-7598-3045-9 (ISBN)
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No detailed description available for "An Introduction to Linear Algebra".
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Edition
Digital original
Language
English
Place of publication
Les Ulis
France
Target group
Professional and scholarly
US School Grade: College Graduate Student
Edition type
Digital original
ISBN-13
978-2-7598-3045-9 (9782759830459)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
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Persons
LIUXuan:
Xuan LIU is a PhD student in mathematics at the University of Macau. are research interests focuses on scientific calculus and numerical linear algebra.
ZHAOZhi:
Zhi ZHAO is Associate Professor of Mathematics at Hangzhou Dianzi University. His research has centered on numerical linear algebra and Riemannian optimization.
LIUWei-Hui:
Wei-Hui LIU is a PhD student in mathematics at the University of Macau. are research interests focuses on scientific calculus and numerical linear algebra.
JINXiao-Qing:
Xiao-Qing JIN is Distinguished Professor of Mathematics at the University of Macau. His research interests include scientific computing, numerical linear algebra, optimization, and financial mathematics.
Xuan LIU is a PhD student in mathematics at the University of Macau. are research interests focuses on scientific calculus and numerical linear algebra.
ZHAOZhi:
Zhi ZHAO is Associate Professor of Mathematics at Hangzhou Dianzi University. His research has centered on numerical linear algebra and Riemannian optimization.
LIUWei-Hui:
Wei-Hui LIU is a PhD student in mathematics at the University of Macau. are research interests focuses on scientific calculus and numerical linear algebra.
JINXiao-Qing:
Xiao-Qing JIN is Distinguished Professor of Mathematics at the University of Macau. His research interests include scientific computing, numerical linear algebra, optimization, and financial mathematics.
Content
- Cover-1
- Cover-2
- An Introduction to Linear Algebra
- Preface
- Contents
- Chapter 1 Linear Systems and Matrices
- 1.1 Introduction to Linear Systems and Matrices
- 1.1.1 Linear equations and linear systems
- 1.1.2 Matrices
- 1.1.3 Elementary row operations
- 1.2 Gauss-Jordan Elimination
- 1.2.1 Reduced row-echelon form
- 1.2.2 Gauss-Jordan elimination
- 1.2.3 Homogeneous linear systems
- 1.3 Matrix Operations
- 1.3.1 Operations on matrices
- 1.3.2 Partition of matrices
- 1.3.3 Matrix product by columns and by rows
- 1.3.4 Matrix product of partitioned matrices
- 1.3.5 Matrix form of a linear system
- 1.3.6 Transpose and trace of a matrix
- 1.4 Rules of Matrix Operations and Inverses
- 1.4.1 Basic properties of matrix operations
- 1.4.2 Identity matrix and zero matrix
- 1.4.3 Inverse of a matrix
- 1.4.4 Powers of a matrix
- 1.5 Elementary Matrices and a Method for Finding A-1
- 1.5.1 Elementary matrices and their properties
- 1.5.2 Main theorem of invertibility
- 1.5.3 A method for finding A-1
- 1.6 Further Results on Systems and Invertibility
- 1.6.1 A basic theorem
- 1.6.2 Properties of invertible matrices
- 1.7 Some Special Matrices
- 1.7.1 Diagonal and triangular matrices
- 1.7.2 Symmetric matrix
- Exercises
- Chapter 2 Determinants
- 2.1 Determinant Function
- 2.1.1 Permutation, inversion, and elementary product
- 2.1.2 Definition of determinant function
- 2.2 Evaluation of Determinants
- 2.2.1 Elementary theorems
- 2.2.2 A method for evaluating determinants
- 2.3 Properties of Determinants
- 2.3.1 Basic properties
- 2.3.2 Determinant of a matrix product
- 2.3.3 Summary
- 2.4 Cofactor Expansions and Cramer's Rule
- 2.4.1 Cofactors
- 2.4.2 Cofactor expansions
- 2.4.3 Adjoint of a matrix
- 2.4.4 Cramer's rule
- Exercises
- Chapter 3 Euclidean Vector Spaces
- 3.1 Euclidean n-Space
- 3.1.1 n-vector space
- 3.1.2 Euclidean n-space
- 3.1.3 Norm, distance, angle, and orthogonality
- 3.1.4 Some remarks
- 3.2 Linear Transformations from Rn to Rm
- 3.2.1 Linear transformations from Rn to Rm
- 3.2.2 Some important linear transformations
- 3.2.3 Compositions of linear transformations
- 3.3 Properties of Transformations
- 3.3.1 Linearity conditions
- 3.3.2 Example
- 3.3.3 One-to-one transformations
- 3.3.4 Summary
- Exercises
- Chapter 4 General Vector Spaces
- 4.1 Real Vector Spaces
- 4.1.1 Vector space axioms
- 4.1.2 Some properties
- 4.2 Subspaces
- 4.2.1 Definition of subspace
- 4.2.2 Linear combinations
- 4.3 Linear Independence
- 4.3.1 Linear independence and linear dependence
- 4.3.2 Some theorems
- 4.4 Basis and Dimension
- 4.4.1 Basis for vector space
- 4.4.2 Coordinates
- 4.4.3 Dimension
- 4.4.4 Some fundamental theorems
- 4.4.5 Dimension theorem for subspaces
- 4.5 Row Space, Column Space, and Nullspace
- 4.5.1 Definition of row space, column space, and nullspace
- 4.5.2 Relation between solutions of Ax = 0 and Ax = b
- 4.5.3 Bases for three spaces
- 4.5.4 A procedure for finding a basis for span(S)
- 4.6 Rank and Nullity
- 4.6.1 Rank and nullity
- 4.6.2 Rank for matrix operations
- 4.6.3 Consistency theorems
- 4.6.4 Summary
- Exercises
- Chapter 5 Inner Product Spaces
- 5.1 Inner Products
- 5.1.1 General inner products
- 5.1.2 Examples
- 5.2 Angle and Orthogonality
- 5.2.1 Angle between two vectors and orthogonality
- 5.2.2 Properties of length, distance, and orthogonality
- 5.2.3 Complement
- 5.3 Orthogonal Bases and Gram-Schmidt Process
- 5.3.1 Orthogonal and orthonormal bases
- 5.3.2 Projection theorem
- 5.3.3 Gram-Schmidt process
- 5.3.4 QR-decomposition
- 5.4 Best Approximation and Least Squares
- 5.4.1 Orthogonal projections viewed as approximations
- 5.4.2 Least squares solutions of linear systems
- 5.4.3 Uniqueness of least squares solutions
- 5.5 Orthogonal Matrices and Change of Basis
- 5.5.1 Orthogonal matrices
- 5.5.2 Change of basis
- Exercises
- Chapter 6 Eigenvalues and Eigenvectors
- 6.1 Eigenvalues and Eigenvectors
- 6.1.1 Introduction to eigenvalues and eigenvectors
- 6.1.2 Two theorems concerned with eigenvalues
- 6.1.3 Bases for eigenspaces
- 6.2 Diagonalization
- 6.2.1 Diagonalization problem
- 6.2.2 Procedure for diagonalization
- 6.2.3 Two theorems concerned with diagonalization
- 6.3 Orthogonal Diagonalization
- 6.4 Jordan Decomposition Theorem
- Exercises
- Chapter 7 Linear Transformations
- 7.1 General Linear Transformations
- 7.1.1 Introduction to linear transformations
- 7.1.2 Compositions
- 7.2 Kernel and Range
- 7.2.1 Kernel and range
- 7.2.2 Rank and nullity
- 7.2.3 Dimension theorem for linear transformations
- 7.3 Inverse Linear Transformations
- 7.3.1 One-to-one and onto linear transformations
- 7.3.2 Inverse linear transformations
- 7.4 Matrices of General Linear Transformations
- 7.4.1 Matrices of linear transformations
- 7.4.2 Matrices of compositions and inverse transformations
- 7.5 Similarity
- Exercises
- Chapter 8 Additional Topics
- 8.1 Quadratic Forms
- 8.1.1 Introduction to quadratic forms
- 8.1.2 Constrained extremum problem
- 8.1.3 Positive definite matrix
- 8.2 Three Theorems for Symmetric Matrices
- 8.3 Complex Inner Product Spaces
- 8.3.1 Complex numbers
- 8.3.2 Complex inner product spaces
- 8.4 Hermitian Matrices and Unitary Matrices
- 8.5 Böttcher-Wenzel Conjecture
- 8.5.1 Introduction
- 8.5.2 Proof of the Böttcher-Wenzel conjecture
- Exercises
- Appendix A Independence of Axioms
- Bibliography
- Index
