Linear Algebra
- Length: 284 pages
- Edition: 1
- Language: English
- Publisher: Routledge
- Publication Date: 2021-03-05
- ISBN-10: 0367697386
- ISBN-13: 9780367697389
- Sales Rank: #4620641 (See Top 100 Books)
This book is intended for a first linear algebra course. The text includes all essential topics in a concise manner and can therefore be fully covered in a one term course. After this course, the student is fully equipped to specialize further in their direction(s) of choice (advanced pure linear algebra, numerical linear algebra, optimization, multivariate statistics, or one of the many other areas of linear algebra applications).
Linear Algebra is an exciting area of mathematics that is gaining more and more importance as the world is becoming increasingly digital. It has the following very appealing features:
- It is a solid axiomatic based mathematical theory that is accessible to a large variety of students.
- It has a multitude of applications from many different fields, ranging from traditional science and engineering applications to more ‘daily life’ applications (internet searches, guessing consumer preferences, etc.).
- It easily allows for numerical experimentation through the use of a variety of readily available software (both commercial and open source).
This book incorporates all these aspects throughout the whole text with the intended effect that each student can find their own niche in the field.
Several suggestions of different software are made. While MATLAB is certainly still a favorite choice, open source programs such as Sage (especially among algebraists) and the Python libraries are increasingly popular. This text guides the student through different programs by providing specific commands.
Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Preface to the Instructor Preface to the Student Acknowledgements Notation List of Figures 1 Matrices and Vectors 1.1 Matrices and Linear Systems 1.2 Row Reduction: Three Elementary Row Operations 1.3 Vectors in ℝn, linear combinations and span 1.4 Matrix Vector Product and the Equation Ax = b 1.5 How to Check Your Work 1.6 Exercises 2 Subspaces in ℝn, Basis and Dimension 2.1 Subspaces in ℝn 2.2 Column Space, Row Space and Null Space of a Matrix 2.3 Linear Independence 2.4 Basis 2.5 Coordinate Systems 2.6 Exercises 3 Matrix Algebra 3.1 Matrix Addition and Multiplication 3.2 Transpose 3.3 Inverse 3.4 Elementary Matrices 3.5 Block Matrices 3.6 Lower and upper triangular matrices and LU factorization 3.7 Exercises 4 Determinants 4.1 Definition of the Determinant and Properties 4.2 Alternative Definition and Proofs of Properties 4.3 Cramer's Rule 4.4 Determinants and Volumes 4.5 Exercises 5 Vector Spaces 5.1 Definition of a Vector Space 5.2 Main Examples 5.3 Linear Independence, Span, and Basis 5.4 Coordinate Systems 5.5 Exercises 6 Linear Transformations 6.1 Definition of a Linear Transformation 6.2 Range and Kernel of Linear Transformations 6.3 Matrix Representations of Linear Transformations 6.4 Change of Basis 6.5 Exercises 7 Eigenvectors and Eigenvalues 7.1 Eigenvectors and Eigenvalues 7.2 Similarity and Diagonalizability 7.3 Complex Eigenvalues 7.4 Systems of Differential Equations: the Diagonalizable Case 7.5 Exercises 8 Orthogonality 8.1 Dot Product and the Euclidean Norm 8.2 Orthogonality and Distance to Subspaces 8.3 Orthonormal Bases and Gram--Schmidt 8.4 Isometries, Unitary Matrices and QR Factorization 8.5 Least Squares Solution and Curve Fitting 8.6 Real Symmetric and Hermitian Matrices 8.7 Singular Value Decomposition 8.8 Exercises Answers to Selected Exercises Appendix A.1 Some Thoughts on Writing Proofs A.1.1 Non-Mathematical Examples A.1.2 Mathematical Examples A.1.3 Truth Tables A.1.4 Quantifiers and Negation of Statements A.1.5 Proof by induction A.1.6 Some Final Thoughts A.2 Complex Numbers A.3 The Field Axioms Index
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