Introductory Linear Algebra
Edition: 1
Copyright: 2021
Pages: 352
Edition: 1
Copyright: 2021
Pages: 352
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Introductory Linear Algebra provides a clear and systematic introduction to the fundamental concepts of linear algebra, emphasizing both theoretical understanding and practical problem-solving. Beginning with vectors and Euclidean spaces, the text develops essential topics including systems of linear equations, matrix algebra, determinants, subspaces, bases, and linear transformations. Through carefully organized chapters and numerous examples, students build the mathematical skills needed to analyze vector spaces, solve linear systems, and understand the structure of matrices and linear mappings.
As the course progresses, readers explore advanced topics such as eigenvalues, eigenvectors, and matrix diagonalization, which form the foundation for many applications in mathematics, engineering, computer science, physics, and data science. Answer keys to selected exercises and suggested readings provide additional opportunities for practice and independent study. By combining geometric intuition with algebraic techniques, this textbook prepares students for further study in higher-level mathematics while demonstrating the broad applicability of linear algebra across STEM disciplines.
Preface
Notations and Conventions
Chapter 1 Euclidean Spaces
1.1 Vectors
1.2 Lines in R2
1.3 Length and Dot Product
1.4 Orthogonal Projection
1.5 Area in R2 and 2×2 Determinants
1.6 Planes in R3
Chapter 2 System of Linear Equations
2.1 Terminologies and Definitions
2.2 Gaussian Elimination
Chapter 3 Matrix Algebra
3.1 Definitions and Properties of Matrix Operations
3.2 Linear Systems Revisited
3.3 Invertible Matrix
3.4 Square Matrices of Special Forms
3.5 Elementary Matrices
Chapter 4 Determinants
4.1 Definition
4.2 Properties of Determinants
4.3 Adjoint Matrix and Cramer’s Rule
4.4 Cross Product in R3
Chapter 5 Subspaces of Rn and Their Bases
5.1 Subspaces of Rn
5.2 Linear Combination and Linear Independence
5.3 Basis and Dimension
5.4 Coordinates with Respect to Ordered Bases
Chapter 6 Linear Transformations
6.1 Matrix Transformations
6.2 Linear Operators on R2 and R3
Chapter 7 Eigenvalues, Eigenvectors and Diagonalization
7.1 Definitions and Properties of Eigenvalues and Eigenvectors
7.2 Diagonalizability
7.3 Diagonalization
Answer Keys to Selected Exercise Problems
Suggested Further Readings
Index
PhD in Mathematics – University of Wisconsin–Madison, 2003
BSc in Mathematics – Sichuan University, 1995
Shengda Hu is a Canadian mathematician and professor specializing in algebraic geometry, symplectic topology, generalized geometry, and related areas of mathematical physics.
I earned my DPhil in Mathematics from the University of Oxford, England, in 1988 and my BSc in Mathematics from Jilin University, China, in 1982.
Prior to joining Laurier, I was a professor in mathematics in Eastern Mediterranean University, North Cyprus, and also worked in the Chinese Academy of Sciences, Beijing, and Atlim University and Cankaya University, Ankara.
I received my PhD and MSc in Mathematics from Chinese Academy of Sciences in 1991 and 1988.
Prior to joining Laurier, I was a professor at the Institute of Mathematics, Chinese Academy of Sciences (1999-2013), and an associate professor at the same institute (1994-1999).
Introductory Linear Algebra provides a clear and systematic introduction to the fundamental concepts of linear algebra, emphasizing both theoretical understanding and practical problem-solving. Beginning with vectors and Euclidean spaces, the text develops essential topics including systems of linear equations, matrix algebra, determinants, subspaces, bases, and linear transformations. Through carefully organized chapters and numerous examples, students build the mathematical skills needed to analyze vector spaces, solve linear systems, and understand the structure of matrices and linear mappings.
As the course progresses, readers explore advanced topics such as eigenvalues, eigenvectors, and matrix diagonalization, which form the foundation for many applications in mathematics, engineering, computer science, physics, and data science. Answer keys to selected exercises and suggested readings provide additional opportunities for practice and independent study. By combining geometric intuition with algebraic techniques, this textbook prepares students for further study in higher-level mathematics while demonstrating the broad applicability of linear algebra across STEM disciplines.
Preface
Notations and Conventions
Chapter 1 Euclidean Spaces
1.1 Vectors
1.2 Lines in R2
1.3 Length and Dot Product
1.4 Orthogonal Projection
1.5 Area in R2 and 2×2 Determinants
1.6 Planes in R3
Chapter 2 System of Linear Equations
2.1 Terminologies and Definitions
2.2 Gaussian Elimination
Chapter 3 Matrix Algebra
3.1 Definitions and Properties of Matrix Operations
3.2 Linear Systems Revisited
3.3 Invertible Matrix
3.4 Square Matrices of Special Forms
3.5 Elementary Matrices
Chapter 4 Determinants
4.1 Definition
4.2 Properties of Determinants
4.3 Adjoint Matrix and Cramer’s Rule
4.4 Cross Product in R3
Chapter 5 Subspaces of Rn and Their Bases
5.1 Subspaces of Rn
5.2 Linear Combination and Linear Independence
5.3 Basis and Dimension
5.4 Coordinates with Respect to Ordered Bases
Chapter 6 Linear Transformations
6.1 Matrix Transformations
6.2 Linear Operators on R2 and R3
Chapter 7 Eigenvalues, Eigenvectors and Diagonalization
7.1 Definitions and Properties of Eigenvalues and Eigenvectors
7.2 Diagonalizability
7.3 Diagonalization
Answer Keys to Selected Exercise Problems
Suggested Further Readings
Index
PhD in Mathematics – University of Wisconsin–Madison, 2003
BSc in Mathematics – Sichuan University, 1995
Shengda Hu is a Canadian mathematician and professor specializing in algebraic geometry, symplectic topology, generalized geometry, and related areas of mathematical physics.
I earned my DPhil in Mathematics from the University of Oxford, England, in 1988 and my BSc in Mathematics from Jilin University, China, in 1982.
Prior to joining Laurier, I was a professor in mathematics in Eastern Mediterranean University, North Cyprus, and also worked in the Chinese Academy of Sciences, Beijing, and Atlim University and Cankaya University, Ankara.
I received my PhD and MSc in Mathematics from Chinese Academy of Sciences in 1991 and 1988.
Prior to joining Laurier, I was a professor at the Institute of Mathematics, Chinese Academy of Sciences (1999-2013), and an associate professor at the same institute (1994-1999).