Linear Algebra
Edition: 1
Copyright: 2021
Pages: 172
Edition: 1
Copyright: 2021
Pages: 172
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This book is intended as a textbook for a one-term undergraduate second linear algebra course which prepares students for both applied mathematics and pure mathematics, in particular, for Galois Theory. The students are assumed to have some knowledge on calculus, linear systems, determinates and matrices. Earlier drafts of this book have been used as the textbook when the author teaches MA222, the second linear algebra course of the three, at Wilfrid Laurier University since 2019.
The book contains 196 exercises of varying difficulty which will allow students to practice their own computational and proof-writing skills. Detailed solutions, answers or hints to all the exercises are provided in the book. Besides standard ones, many of the exercises are provided in the book. Besides standard ones, many of the exercises are interesting. Some are rather hard. It is not a surprise if the reader cannot solve some of the exercises by themselves, particularly for first learners. It is strongly suggested that the readers first try to solve them without looking at the solutions of hints.
Features of this book: This book is written in a way that is easy to understand, simple and concise. This makes the book a small one. The differences of this book from many others are following features.
1. Complex Numbers, Polynomials Over a Field
1.1 Arithmetic of complex numbers
1.2 Polar form of complex numbers
1.3 Polynomials over a field
1.4 Irreducible polynomials
1.5 Exercises
2. Determinants
2.1 Evaluating determinants
2.2 Exercises
3. Vector Spaces and Subspaces
3.1 Basic properties of vector spaces
3.2 Subspaces
3.3 Spanning sets and linear dependence
3.4 Bases and dimension
3.5 Applications and Sum-Dimension Formula
3.6 Quotient spaces
3.7 Exercises
4. Linear Maps
4.1 Linear maps, Dimension Theorem
4.2 The matrix of a linear map
4.3 Composition of linear maps
4.4 Invertible linear maps
4.5 The transition matrix
4.6 Dual spaces and transpose maps
4.7 Exercises
5. The Rank of a Matrix
5.1 The rank of a matrix
5.2 Exercises
6. Diagonalization of Linear Operators
6.1 Eigenvalues and eigenvectors
6.2 Cayley–Hamilton Theorem
6.3 Diagonalizability of linear operators
6.4 Direct sum of subspaces
6.5 Exercises
7. Inner Product Spaces and Bilinear Forms
7.1 Basic properties of inner product spaces
7.2 The simplified Gram–Schmidt orthogonalization
7.3 Orthogonal complements
7.4 Bilinear forms and Sylvester’s law of inertia
7.5 Quadratic forms
7.6 Exercises
Solutions and Hints
Appendix A. Equivalence Relations and Kuratowski–Zorn Lemma
References
Index