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Linear Algebra Done Wrong

Sergei Treil
Department of Mathematics, Brown University

The title of the book sounds a bit mysterious. Why should anyone read this
book if it presents the subject in a wrong way? What is particularly done
"wrong" in the book?
Before answering these questions, let me first describe the target audience
of this text. This book appeared as lecture notes for the course "Honors
Linear Algebra". It supposed to be a first linear algebra course for mathematically
advanced students. It is intended for a student who, while not
yet very familiar with abstract reasoning, is willing to study more rigorous
mathematics that is presented in a "cookbook style" calculus type course.
Besides being a first course in linear algebra it is also supposed to be a
first course introducing a student to rigorous proof, formal definitions|in
short, to the style of modern theoretical (abstract) mathematics. The target
audience explains the very specific blend of elementary ideas and concrete
examples, which are usually presented in introductory linear algebra texts
with more abstract definitions and constructions typical for advanced books.
Another specific of the book is that it is not written by or for an algebraist.
So, I tried to emphasize the topics that are important for analysis,
geometry, probability, etc., and did not include some traditional topics. For
example, I am only considering vector spaces over the fields of real or complex
numbers. Linear spaces over other fields are not considered at all, since
I feel time required to introduce and explain abstract fields would be better
spent on some more classical topics, which will be required in other disciplines.
And later, when the students study general fields in an abstract
algebra course they will understand that many of the constructions studied
in this book will also work for general fields.

Also, I treat only finite-dimensional spaces in this book and a basis
always means a finite basis. The reason is that it is impossible to say something
non-trivial about infinite-dimensional spaces without introducing convergence,
norms, completeness etc., i.e. the basics of functional analysis.
And this is definitely a subject for a separate course (text). So, I do not
consider infinite Hamel bases here: they are not needed in most applications
to analysis and geometry, and I feel they belong in an abstract algebra

Lecture Notes

Linear Algebra Done Wrong

Sergei Treil
Department of Mathematics, Brown University

Preface iii

Chapter 1. Basic Notions 1

1. Vector spaces 1
2. Linear combinations, bases. 5
3. Linear Transformations. Matri{vector multiplication 11
4. Composition of linear transformations and matri multiplication. 16
5. Invertible transformations and matrices. Isomorphisms 21
6. Subspaces. 27
7. Application to computer graphics. 28

Chapter 2. Systems of linear equations 35

1. Di erent faces of linear systems. 35
2. Solution of a linear system. Echelon and reduced echelon forms 36
3. Analyzing the pivots. 42
4. Finding A??1 by row reduction. 47
5. Dimension. Finite-dimensional spaces. 49
6. General solution of a linear system. 51
7. Fundamental subspaces of a matri. Rank. 54
8. Representation of a linear transformation in arbitrary bases.
Change of coordinates formula. 62

Chapter 3. Determinants 69

1. Introduction. 69
2. What properties determinant should have. 70
3. Constructing the determinant. 72
4. Formal de nition. Eistence and uniqueness of the determinant. 80
5. Cofactor epansion. 83
6. Minors and rank. 89
7. Review eercises for Chapter 3. 90

Chapter 4. Introduction to spectral theory (eigenvalues andeigenvectors) 93

1. Main de nitions 94
2. Diagonalization. 99

Chapter 5. Inner product spaces 109

1. Inner product in Rn and Cn. Inner product spaces. 109
2. Orthogonality. Orthogonal and orthonormal bases. 117
3. Orthogonal projection and Gram-Schmidt orthogonalization 120
4. Least square solution. Formula for the orthogonal projection 127
5. Fundamental subspaces revisited. 132
6. Isometries and unitary operators. Unitary and orthogonal matrices. 136

Chapter 6. Structure of operators in inner product spaces. 143

1. Upper triangular (Schur) representation of an operator. 143
2. Spectral theorem for self-adjoint and normal operators. 145
3. Polar and singular value decompositions. 150
4. What do singular values tell us? 158
5. Structure of orthogonal matrices 164
6. Orientation 170

Chapter 7. Bilinear and quadratic forms 175

1. Main de nition 175
2. Diagonalization of quadratic forms 177
3. Silvester's Law of Inertia 182
4. Positive de nite forms. Minima characterization of eigenvalues
and the Silvester's criterion of positivity 184

Chapter 8. Advanced spectral theory 191

1. Cayley{Hamilton Theorem 191
2. Spectral Mapping Theorem 195
Contents i
3. Generalized eigenspaces. Geometric meaning of algebraic
multiplicity 197
4. Structure of nilpotent operators 204
5. Jordan decomposition theorem 210   Tell A Friend