There is good reason to be excited about Linear Algebra. With the world becoming increasingly digital, Linear Algebra is gaining more and more importance. When we send texts, share video, do internet searches, there are Linear Algebra algorithms in the background that make it work.
This concise introduction to Linear Algebra is authored by a leading researcher presents a book that covers all the requisite material for a first course on the topic in a more practical way.
The book focuses on the development of the mathematical theory and presents many applications to assist instructors and students to master the material and apply it to their areas of interest, whether it be to further their studies in mathematics, science, engineering, statistics, economics, or other disciplines.
Linear Algebra has 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.
•It easily allows for numerical experimentation through the use of a variety of readily available software (both commercial and open source).
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 to try out different programs by providing specific commands.
Table of Contents
Preface to the Instructor
Preface to the Student
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 Spaces, Row Spaces 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. Transpose. Inverse. Elementary Matrices. Block Matrices. Lower and Upper Triangular Matrices and LU Factorization 3.7 Exercises
4.Determinants. Definition of the Determinant and Properties. Alternative Definition and Proofs of Properties. Cramer’s rule. Determinants and Volumes. Exercises
5.Vector spaces. Definition of a Vector Space. Main Examples. Linear Independence, Span, and Basis. Coordinate Systems. Exercises.
6.Linear Transformations. Definition of a Linear Transformation. Range and Kernel of Linear Transformations. Matrix Representations of Linear Maps. Change of Basis. Exercises.
7.Eigenvectors and Eigenvalues. Eigenvectors and Eigenvalues. Similarity and Diagonalizability. Complex eigenvalues. Exercises.
8.Orthogonality. Dot Product and the Euclidean Norm. Orthogonality and Distance to Subspaces. Orthonormal Bases and Gram-Schmidt. Unitary Matrices and QR Factorization. Least Squares Solution and Curve Fitting. Real Symmetric and Hermitian Matrices.
Appendix.A1.1 Some Thoughts on Writing Proofs. A1.2 Mathematical Examples. A 1.3 Truth Tables. A1.4 Quantifiers and Negation of Statements. A1.5 Proof by Induction. A1.6 Some Final Thoughts. A.2 Complex numbers. A.3 The Field Axioms.
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.