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.
| Preface |
|
xi | |
| Preface to the Instructor |
|
xi | |
| Preface to the Student |
|
xiv | |
| Acknowledgments |
|
xvii | |
| Notation |
|
xix | |
|
|
|
xxi | |
|
|
|
1 | (32) |
|
1.1 Matrices and Linear Systems |
|
|
2 | (2) |
|
1.2 Row Reduction: Three Elementary Row Operations |
|
|
4 | (7) |
|
1.3 Vectors in Rn, Linear Combinations and Span |
|
|
11 | (6) |
|
1.4 Matrix Vector Product and the Equation Ax = b |
|
|
17 | (5) |
|
1.5 How to Check Your Work |
|
|
22 | (1) |
|
|
|
23 | (10) |
|
2 Subspaces in Rn, Basis and Dimension |
|
|
33 | (28) |
|
|
|
33 | (3) |
|
2.2 Column Space, Row Space and Null Space of a Matrix |
|
|
36 | (3) |
|
|
|
39 | (3) |
|
|
|
42 | (5) |
|
|
|
47 | (3) |
|
|
|
50 | (11) |
|
|
|
61 | (28) |
|
3.1 Matrix Addition and Multiplication |
|
|
61 | (4) |
|
|
|
65 | (1) |
|
|
|
66 | (3) |
|
|
|
69 | (2) |
|
|
|
71 | (4) |
|
3.6 Lower and Upper Triangular Matrices and LU Factorization |
|
|
75 | (4) |
|
|
|
79 | (10) |
|
|
|
89 | (22) |
|
4.1 Definition of the Determinant and Properties |
|
|
89 | (3) |
|
4.2 Alternative Definition and Proofs of Properties |
|
|
92 | (7) |
|
|
|
99 | (2) |
|
4.4 Determinants and Volumes |
|
|
101 | (2) |
|
|
|
103 | (8) |
|
|
|
111 | (24) |
|
5.1 Definition of a Vector Space |
|
|
111 | (2) |
|
|
|
113 | (2) |
|
5.3 Linear Independence, Span, and Basis |
|
|
115 | (8) |
|
|
|
123 | (3) |
|
|
|
126 | (9) |
|
|
|
135 | (26) |
|
6.1 Definition of a Linear Transformation |
|
|
135 | (5) |
|
6.2 Range and Kernel of Linear Transformations |
|
|
140 | (5) |
|
6.3 Matrix Representations of Linear Transformations |
|
|
145 | (2) |
|
|
|
147 | (2) |
|
|
|
149 | (12) |
|
7 Eigenvectors and Eigenvalues |
|
|
161 | (20) |
|
7.1 Eigenvectors and Eigenvalues |
|
|
161 | (3) |
|
7.2 Similarity and Diagonalizability |
|
|
164 | (2) |
|
|
|
166 | (2) |
|
7.4 Systems of Differential Equations: the Diagonalizable Case |
|
|
168 | (3) |
|
|
|
171 | (10) |
|
|
|
181 | (62) |
|
8.1 Dot Product and the Euclidean Norm |
|
|
181 | (5) |
|
8.2 Orthogonality and Distance to Subspaces |
|
|
186 | (7) |
|
8.3 Orthonormal Bases and Gram-Schmidt |
|
|
193 | (5) |
|
8.4 Isometries, Unitary Matrices and QR Factorization |
|
|
198 | (5) |
|
8.5 Least Squares Solution and Curve Fitting |
|
|
203 | (3) |
|
8.6 Real Symmetric and Hermitian Matrices |
|
|
206 | (3) |
|
8.7 Singular Value Decomposition |
|
|
209 | (8) |
|
|
|
217 | (18) |
|
Answers to Selected Exercises |
|
|
235 | (8) |
|
|
|
243 | (16) |
|
A.1 Some Thoughts on Writing Proofs |
|
|
243 | (9) |
|
A.1.1 Non-Mathematical Examples |
|
|
243 | (2) |
|
A.1.2 Mathematical Examples |
|
|
245 | (2) |
|
|
|
247 | (1) |
|
A.1.4 Quantifiers and Negation of Statements |
|
|
248 | (3) |
|
|
|
251 | (1) |
|
A.1.6 Some Final Thoughts |
|
|
252 | (1) |
|
|
|
252 | (4) |
|
|
|
256 | (3) |
| Index |
|
259 | |
Hugo J. Woerdeman, PhD, professor, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, USA, is also the author of Advanced Linear Algebra, published by CRC Press, and a co-author of Matrix Completions, Moments, and Sums of Squares, published by Princeton University Press. He also serves as Vice President of two societies of researchers: The International Linear Algebra Society and The International Workshop on Operator Theory and its Applications.
