The Oxford Linear Algebra for Scientists

cover
titlepage
copyright
dedication
preface
Acknowledgements
contents
Linearity — an informal introduction
	Why linearity?
	Linearity, more abstractly
		Linear functions
		Linear equations
		Vectors with two components
		Linearity for maps between vectors
		Linear maps and matrices
		Back to linear equations
	Plan of the book
	Exercises
Part I Preliminaries
	Sets and functions
		Sets
			(Non-) definition of sets
			Set operations
			New sets from old ones
		Relations
			Basic definitions
			Properties of equivalence relations
		Functions
			Definition of functions
			Composition of functions
			Properties of functions
			The inverse function
		Rudiments of logic
			Predicates and Boolean operations
			Implications
			Quantifiers
			Patterns of proofs
		Exercises
	Groups
		Definition and basic properties
			Definition
			Examples of groups
			Sub-groups
			Group homomorphisms
		Permutation groups
			Calculating with permutations
			Permutations in terms of transpositions
			The sign of permutations
		Exercises
	Fields
		Fields and their properties
			Definition
			Some conclusions from the field axioms
			Order on fields
		Examples of fields
		The complex numbers
			Construction of complex numbers
			Complex conjugation
			Beyond R2
		Basics of polynomials
			Basics and polynomial division
			Zeros and multiplicity
			Factorization
		Exercises
Part II Vector spaces
	Coordinate vectors
		Basic definitions
			Definition of coordinate vectors
			Addition and scalar multiplication
			Calculating with coordinate vectors
		Standard unit vectors
			Definition of standard unit vectors
			Calculating with standard unit vectors
		Exercises
	Vector spaces
		Basic definitions
			Vector space axioms
			Implications of vector space axioms
			Vector subspaces
			Linear Maps
			Algebras
		Examples of vector spaces
			Coordinate vector spaces
			Matrices and matrix vector spaces
			Vector spaces of functions
		Exercises
	Elementary vector space properties
		Linear independence
			Linear combinations and span
			Linear independence
			Properties of linearly independent vectors
			Examples for linear independence
		Basis and dimension
			Basis and coordinates
			Examples of bases and coordinates
			Dimension of a vector space
			Existence of a basis
			Properties of finite-dimensional vector spaces
		Exercises
	Vector subspaces
		Intersection and sum
			Intersection of vector subspaces
			Union and sum
			Dimension of vector space sums
			Direct sums
			Direct sums of vector spaces
		Quotient spaces*
			Equivalence relation and cosets
			Quotient vector space
		Exercises
Part III Basic geometry
	The dot product
		Basic properties
			Definition of dot product
			Properties of the dot product
		Length and angle
			Definition of length
			The Cauchy–Schwarz inequality
			Properties of the length
			The angle between vectors
		Orthogonality
			Definition of orthogonality
			The Kronecker delta symbol
			Orthonormal basis
		Exercises
	Vector and triple product
		The cross product
			Orthogonality in two dimensions
			Definition of cross product in R3
			Existence and uniqueness of the cross product
			The Levi-Civita symbol in R3
			Properties of the cross product
			Geometrical interpretation of the cross product
		The triple product
			Definition of triple product
			Calculation of the determinant
			Interpretation of the triple product
		Exercises
	Lines and planes
		Lines in R2
			Parametric and Cartesian form
			Intersection of two lines
		Lines and planes in R3
			Parametric and Cartesian form for planes
			Parametric and Cartesian form for lines
			Minimal distances
			Intersection of two planes
			Intersection of line and plane
			Intersection of three planes
		Exercises
Part IV Linear maps and matrices
	Introduction to linear maps
		First properties of linear maps
			Reminder of definition
			Existence and construction of linear maps
			Addition and scalar multiplication of linear maps
			Map composition and inverse
			Isomorphisms and general linear groups
		Examples of linear maps
			Coordinate maps
			Differential operators
		Exercises
	Matrices
		Matrices as linear maps
			Linear maps between coordinate vectors
			Matrix-vector multiplication
			The two faces of matrices
			Square and diagonal matrices
		Matrix multiplication
			Matrix multiplication from map composition
			Rules for matrix multiplication
			Matrix inverse and general linear group
		Transpose and Hermitian conjugate
			The transpose of a matrix
			Symmetric and anti-symmetric matrices
			Properties of transposition
			The Hermitian conjugate of a matrix
			Hermitian and anti-Hermitian matrices
			Properties of Hermitian conjugation
		Exercises
	The structure of linear maps
		Image and kernel
			Definition of image and kernel
			Rank of a linear map
			Injective and surjective linear maps
		The rank theorem
			Motivation
			The theorem
			Easy conclusions from the rank theorem
			Isomorphisms
			The inverse of a linear map
		Another proof of the rank theorem*
		Exercises
	Linear maps in terms of matrices
		Matrices representing linear maps
			Basis choice
			Computing the representing matrix
			Examples for matrices describing linear maps
		Change of basis
			General case
			Identical domain and co-domain
			Conjugate matrices
		Exercises
Part V Linear systems and algorithms
	Computing with matrices
		Row operations
			Definition of row operations
			Upper echelon form
			Algorithm to bring a matrix into upper echelon form
		Rank of a matrix
			Row and column rank
			Computing the rank
		Matrix inverse
			The elementary matrices
			Algorithm to calculate the matrix inverse
		Exercises
	Linear systems
		Abstract linear systems
			Definition of linear systems
			Structure of solution space
		Systems of linear equations
			Definition
			Solutions of homogeneous system
			Solution of inhomogeneous system
			Examples with explicit calculation
			Row operations for linear equations
			Algorithm for solving linear equations
		Applications to geometry
			Parametric and Cartesian form
			Intersection of affine k-planes
			Intersections and linear systems
		Exercises
	Determinants
		Existence and uniqueness
			Definition of determinant
			The general formula for the determinant
			The Levi-Civita symbol
			The determinant in low dimensions
			Determinants for triangular matrices
		Properties of the determinant
			Determinant and transposition
			Determinant and matrix multiplication
			Determinant and basis transformation
			Orientation
		Computing with determinants
			The co-factor matrix
			Laplace expansion of determinant
			Matrix inverse from co-factors
			Determinant and row operations
			Minors
			Cramer's rule
		Exercises
Part VI Eigenvalues and eigenvectors
	Basics of eigenvalues
		Eigenvalues and eigenspaces
			Definition of eigenvalues and eigenvectors
			Degeneracy and eigenspaces
		The characteristic polynomial
			Definition of characteristic polynomial
			Properties of the characteristic polynomial
			Examples
			Degeneracy and multiplicity
			Class functions
		The theorem of Cayley–Hamilton*
			Polyomials of endomorphisms
			The minimal polynomial
			The theorem
		Exercises
	Diagonalizing linear maps
		Diagonalization
			Basic criteria
			The diagonal matrix
			Diagonalizing and class functions
		Examples
		Projectors
			Definition of projectors
			Diagonalizing projectors
		Simultaneous diagonalization*
			Diagonalization of restricted maps
			Criterion for simultaneous diagonalization
		Exercises
	The Jordan normal form*
		Nilpotent endormorphisms*
			Powers of endomorphisms
			Definition of nilpotentency
			Structure of nilpotent endomorphisms
			Examples
		The Jordan form*
			The decomposition theorem
			The theorem
			Implications of Jordan normal form
		Examples*
		Exercises
Part VII Inner product vector spaces
	Scalar products
		Real and Hermitian scalar products
			Definition of norms
			Definition of scalar products
			The norm associated to a scalar product
			Orthogonal vectors and angles
		Examples of scalar products
		Orthogonality and Gram–Schmidt procedure
			Ortho-normal bases
			Existence of ortho-normal bases
			Construction of ortho-normal bases
			Properties of ortho-normal bases
			Orthogonal spaces
		Exercises
	Adjoint and unitary maps
		Adjoint and self-adjoint maps
			Definition and basic properties of adjoint map
			Adjoint map relative to a basis
			Examples
			Kernel and image of the adjoint map
			Self-adjoint maps
		Unitary maps
			Definition of unitary maps
			Unitary groups
			Orthogonal matrices
			Unitary matrices
		Exercises
	Diagonalization — again
		Hermitian maps
			Eigenvectors and eigenvalues of Hermitian maps
			Diagonalizing Hermitian maps
			Examples
		Normal maps*
			Definition of normal maps
			Diagonalization of normal maps
			Diagonalizing unitary maps
			Orthogonal matrices
			Three-dimensional rotations — again
		Singular value decomposition*
			General bases
			Ortho-normal bases
		Functions of matrices*
			Defining functions of matrices
			Matrix functions and diagonalization
			Direct computation of matrix functions
		Exercises
	Bi-linear and sesqui-linear forms*
		Basics definitions*
			Definition of bi-linear and sesqui-linear forms
			The associated quadratic form
			Linear forms relative to a basis
			Positive definiteness
			Degeneracy
		Classification of linear forms*
			A normal form for the describing matrix
			Theorem of Sylvester
			Groups associated to linear forms
		Quadratic hyper-surfaces*
			Definition of quadratic hyper-surfaces
			Diagonalization of quadratic hyper-surfaces
			Quadratic curves in R2
			Quadratic surfaces in R3
		Exercises
Part VIII Dual and tensor vector spaces*
	The dual vector space*
		Definition of dual vector space*
			Linear functionals
			Dual basis
			Index notation
			The double dual
		The dual map*
			The orthogonal space
			The dual map
			Kernel and image of the dual map
		Linear forms and dual space*
			The map between V and V*
			Index notation — again
		Exercises
	Tensors*
		Tensor basics*
			Definition of tensors
			The tensor product
			The universal property
			Indices
			Basis transformation of tensors
			Induced maps on tensors
		Further tensor properties*
			Symmetric and anti-symmetric tensors
			Linear maps as tensors
			Bi-linear forms and tensors
		Multi-linearity*
			Higher-rank tensors
			(p,q) tensors
			Alternating q forms
			The determinant as an alternating form
			The outer algebra of R3
		Exercises
	References
	Index