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Linear algebra is the study of vector spaces and the linear maps between them. It underlies much of modern mathematics and is widely used in applications.

A (Terse) Introduction to Linear Algebrais a concise presentation of the core material of the subject—those elements of linear algebra that every mathematician, and everyone who uses mathematics, should know. It goes from the notion of a finite-dimensional vector space to the canonical forms of linear operators and their matrices, and covers along the way such key topics as: systems of linear equations, linear operators and matrices, determinants, duality, and the spectral theory of operators on inner-product spaces.

The last chapter offers a selection of additional topics indicating directions in which the core material can be applied. The Appendix provides all the relevant background material.

Written for students with some mathematical maturity and an interest in abstraction and formal reasoning, the book is self-contained and is appropriate for an advanced undergraduate course in linear algebra.

Yitzhak Katznelson, Stanford University, Stanford, CA,
Yonatan R. Katznelson, University of California, Santa Cruz, Santa Cruz, CA

Preface           10
1. Vector Spaces           12
1.1 Groups and fields           12
1.2 Vector spaces           15
1.3 Linear dependence, bases, and dimension           25
1.4 Systems of linear equations           33
*1.5 Normed finite-dimensional linear spaces           43

2. Linear Operators and Matrices           46
2.1 Linear operators           46
2.2 Operator multiplication           50
2.3 Matrix multiplication           52
2.4 Matrices and operators           57
2.5 Kernel, range, nullity, and rank           62

*2.6 Operator norms           67
3. Duality of Vector Spaces           68
3.1 Linear functionals           68
3.2 The adjoint           73

4. Determinants           76
4.1 Permutations           76
4.2 Multilinear maps           80
4.3 Alternating n-forms           85
4.4 Determinant of an operator           87
4.5 Determinant of a matrix           90

5. Invariant Subspaces           96
5.1 The characteristic polynomial           96
5.2 Invariant subspaces           99
5.3 The minimal polynomial           104

6. Inner-Product Spaces           114
6.1 Inner products           114
6.2 Duality and the adjoint           122
6.3 Self-adjoint operators           124
6.4 Normal operators           130
6.5 Unitary and orthogonal operators           132
*6.6 Positive definite operators           138
*6.7 Polar decomposition           139
*6.8 Contractions and unitary dilations           143

7. Structure Theorems           146
7.1 Reducing subspaces           146
7.2 Semisimple systems           153
7.3 Nilpotent operators           158
7.4 The Jordan canonical form           162
*7.5 The cyclic decomposition, general case           163
*7.6 The Jordan canonical form, general case           167

8. Additional Topics           170
8.1 Functions of an operator           170
8.2 Quadratic forms           173
8.3 Perron-Frobenius theory           177
8.4 Stochastic matrices           189
8.5 Representation of finite groups           191

A Appendix           198
A.1 Equivalence relations-partitions           198
A.2 Maps           199
A.3 Groups           200
*A.4 Group actions           205
A.5 Rings and algebras 207
A.6 Polynomials           212

• Index           222