Cover
Copyright
Contributors
Table of Contents
Preface
Section 1: Introduction
Chapter 1: Superposition with Euclid
Vectors
Vector addition
Scalar multiplication
Linear combinations
Superposition
Measurement
Summary
Answers to exercises
Exercise 1
Exercise 2
Chapter 2: The Matrix
Defining a matrix
Notation
Redefining vectors
Simple matrix operations
Addition
Scalar multiplication
Transposing a matrix
Defining matrix multiplication
Multiplying vectors
Matrix-vector multiplication
Matrix multiplication
Properties of matrix multiplication
Special types of matrices
Square matrices
Identity matrices
Quantum gates
Logic gates
Circuit model
Summary
Answers to exercises
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
References
Section 2: Elementary Linear Algebra
Chapter 3: Foundations
Sets
The definition of a set
Notation
Important sets of numbers
Tuples
The Cartesian product
Functions
The definition of a function
Exercise 1
Invertible functions
Binary operations
The definition of a binary operation
Properties
Groups
Fields
Exercise 2
Vector space
Summary
Answers to Exercises
Exercise 1
Exercise 2
Works cited
Chapter 4: Vector Spaces
Subspaces
Definition
Examples
Exercise 1
Linear independence
Linear combination
Linear dependence
Span
Basis
Dimension
Summary
Answers to exercises
Exercise 1
Chapter 5: Using Matrices to Transform Space
Linearity
What is a linear transformation?
Describing linear transformations
Representing linear transformations with matrices
Matrices depend on the bases chosen
Matrix multiplication and multiple transformations
The commutator
Transformations inspired by Euclid
Translation
Rotation
Projection
Exercise two
Linear operators
Linear functionals
A change of basis
Summary
Answers to exercises
Exercise one
Exercise two
Works cited
Section 3: Adding Complexity
Chapter 6: Complex Numbers
Three forms, one number
Definition of complex numbers
Cartesian form
Addition
Multiplication
Exercise 1
Complex conjugate
Absolute value or modulus
Division
Powers of i
Polar form
Polar coordinates
Exercise 3
Defining complex numbers in polar form
Example
Multiplication and division in polar form
Example
De Moivre's theorem
The most beautiful equation in mathematics
Exponential form
Exercise 4
Conjugation
Multiplication
Example
Conjugate transpose of a matrix
Bloch sphere
Summary
Exercises
Exercise 1
Exercise 2
Exercise 3
Exercise 4
References
Chapter 7: EigenStuff
The inverse of a matrix
Determinants
Exercise one
The invertible matrix theorem
Calculating the inverse of a matrix
Exercise two
Eigenvalues and eigenvectors
Definition
Example with a matrix
The characteristic equation
Finding eigenvectors
Multiplicity
Trace
The special properties of eigenvalues
Summary
Answers to exercises
Exercise one
Exercise two
Chapter 8: Our Space in the Universe
The inner product
Orthonormality
The norm
Orthogonality
Orthonormal vectors
The Kronecker delta function
The outer product
Exercise two
Operators
Representing an operator using the outer product
Exercise 3
The completeness relation
The adjoint of an operator
Types of operators
Normal operators
Hermitian operators
Unitary operators
Projection operators
Positive operators
Tensor products
The tensor product of vectors
Exercise four
The basis of tensor product space
Exercise five
The tensor product of operators
Exercise six
The inner product of composite vectors
Exercise seven
Summary
Answers to exercises
Exercise one
Exercise two
Exercise three
Exercise four
Exercise five
Exercise six
Exercise seven
Chapter 9: Advanced Concepts
Gram-Schmidt
Cauchy-Schwarz and triangle inequalities
Spectral decomposition
Diagonal matrices
Spectral theory
Example
Bra-ket notation
Example take two
Singular value decomposition
Polar decomposition
Operator functions and the matrix exponential
Summary
Works cited
Section 4: Appendices
Appendix 1: Braβket Notation
Operators
Bras
Appendix 2: Sigma Notation
Sigma
Variations
Summation rules
Appendix 3: Trigonometry
Measuring angles
Degrees
Radians
Trigonometric functions
Formulas
Summary
The trig cheat sheet
Pythagorean identities
Double angle identities
Sum/difference identities
Product-to-sum identities
Works cited
Appendix 4: Probability
Definitions
Random variables
Discrete random variables
The measures of a random variable
Summary
Works cited
Appendix 5: References
Index
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