# Book:Richard Kaye/Linear Algebra

## Richard Kaye and Robert Wilson: Linear Algebra

Published $\text {1998}$, Oxford Science Publications

ISBN 0-19-850237-0.

### Contents

Preface
PART I MATRICES AND VECTOR SPACES
1 Matrices
1.1 Matrices
1.2 Addition and multiplication of matrices
1.3 The inverse of a matrix
1.4 The transpose of a matrix
1.5 Row and column operations
1.6 Determinant and trace
1.7 Minors and cofactors
2 Vector spaces
2.1 Examples and axioms
2.2 Subspaces
2.3 Linear independence
2.4 Bases
2.5 Coordinates
2.6 Vector spaces over other fields

PART II BILINEAR AND SESQUILINEAR FORMS
3 Inner product spaces
3.1 The standard inner product
3.2 Inner products
3.3 Inner products over $\C$
4 Bilinear and sesquilinear forms
4.1 Bilinear forms
4.2 Representation by matrices
4.3 The base-change formula
4.4 Sesquilinear forms over $\C$
5 Orthogonal bases
5.1 Orthonormal bases
5.2 The Gram-Schmidt process
5.3 Properties of orthonormal bases
5.4 Orthogonal complements
6 When is a form definite?
6.1 The Gram-Schmidt process revisited
7 Quadratic forms and Sylvester's law of inertia
7.2 Sylvester's law of inertia
7.3 Examples
7.4 Applications to surfaces
7.5 Sesquilinear and Hermitian forms

PART III LINEAR TRANSFORMATIONS
8 Linear transformations
8.1 Basics
8.2 Arithmetic operations on linear transformations
8.3 Representation by matrices
9 Polynomials
9.1 Polynomials
9.2 Evaluating polynomials
9.3 Roots of polynomials over $\C$
9.4 Roots of polynomials over other fields
10 Eigenvalues and eigenvectors
10.1 An example
10.2 Eigenvalues and eigenvectors
10.3 Upper triangular matrices
11 The minimum polynomial
11.1 The minimum polynomial
11.2 The characteristic polynomial
11.3 The Cayley-Hamilton theorem
12 Diagonalization
12.1 Diagonal matrices
12.2 A criterion for diagonalizability
12.3 Examples
13.1 Orthogonal and unitary transformations
13.2 From forms to transformations
13.3 Eigenvalues and diagonalization
13.4 Applications
14 The Jordan normal form
14.1 Jordan normal form
14.2 Obtaining the Jordan normal form
14.3 Applications
14.4 Proof of the primary decomposition theorem
Appendix A A Theorem of Analysis
Appendix B Applications to quantum mechanics
Index

Next

## Errata

### Simultaneous Linear Equations: Arbitrary System $6$

Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations

To solve:

$\begin {array} {rcrcrcr} x & + & y & + & 2 z & = & -1 \\ -x & + & & & z & = & -1 \\ -x & + & y & + & 4 z & = & 3 \\ \end {array}$,

first put the equation in matrix form

$\paren {\begin {array} {rrr} 1 & 1 & 2 \\ -1 & 0 & 1 \\ -1 & 1 & 4 \end {array} } \begin {pmatrix} x \\ y \\ z \end {pmatrix} = \paren {\begin {array} {r} -1 \\ -1 \\ 3 \end {array} }$

and then put the augmented matrix formed from the matrix on the left with the column vector on the right into echelon form:

$\paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ -1 & 0 & 1 & -1 \\ -1 & 1 & 4 & 3 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 2 & 6 & -4 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 0 & 0 & 0 \end {array} }$.