Linear algebra thoroughly explained

1. Vector Spaces

1.1. Introduction

1.2. Geometrical Vectors in a Plane

1.3. Vectors in a Cartesian (Analytic) Plane R[superscript 2]

1.4. Scalar Multiplication (The Product of a Number with a Vector)

1.5. The Dot Product of Two Vectors (or the Euclidean Inner Product of Two Vectors in R[superscript 2])

1.6. Applications of the Dot Product and Scalar Multiplication

1.7. Vectors in Three-Dimensional Space (Spatial Vectors)

1.8. The Cross Product in R[superscript 3]

1.9. The Mixed Triple Product in R[superscript 3]. Applications of the Cross and Mixed Products

1.10. Equations of Lines in Three-Dimensional Space

1.11. Equations of Planes in Three-Dimensional Space

1.12. Real Vector Spaces and Subspaces

1.13. Linear Dependence and Independence. Spanning Subsets and Bases

1.14. The Three Most Important Examples of Finite-Dimensional Real Vector Spaces

1.14.1. The Vector Space R[superscript n] (Number Columns)

1.14.2. The Vector Space R[subscript n x n] (Matrices)

1.14.3. The Vector Spaces P[subscript 3] (Polynomials)

1.15. Some Special Topics about Matrices

1.15.1. Matrix Multiplication

1.15.2. Some Special Matrices

A. Determinants

A.1. Definitions of Determinants

A.2. Properties of Determinants

2. Linear Mappings and Linear Systems

2.1. A Short Plan for the First 5 Sections of Chapter 2

2.2. Some General Statements about Mapping

2.3. The Definition of Linear Mappings (Linmaps)

2.4. The Kernel and the Range of L

2.5. The Quotient Space V[subscript n]/ker L and the Isomorphism V[subscript n]/ker [characters not reproducible] ran L

2.6. Representation Theory

2.6.1. The Vector Space L(V[subscript n], W[subscript m])

2.6.2. The Linear Map M : R[superscript n] to R[superscript m]

2.6.3. The Three Isomorphisms v, w and v - w

2.6.4. How to Calculate the Representing Matrix M

2.7. An Example (Representation of a Linmap Which Acts between Vector Spaces of Polynomials)

2.8. Systems of Linear Equations (Linear Systems)

2.9. The Four Tasks

2.10. The Column Space and the Row Space

2.11. Two Examples of Linear Dependence of Columns and Rows of a Matrix

2.12. Elementary Row Operations (Eros) and Elementary Matrices

2.12.1. Eros

2.12.2. Elementary Matrices

2.13. The GJ Form of a Matrix

2.14. An Example (Preservation of Linear Independence and Dependence in GJ Form)

2.15. The Existence of the Reduced Row-Echelon (GJ) Form for Every Matrix

2.16. The Standard Method for Solving A X = b

2.16.1. When Does a Consistent System A X = b Have a Unique Solution?

2.16.2. When a Consistent System A X = b Has No Unique Solution

2.17. The GJM Procedure - a New Approach to Solving Linear Systems with Nonunique Solutions

2.17.1. Detailed Explanation

2.18. Summary of Methods for Solving Systems of Linear Equations

3. Inner-Product Vector Spaces (Euclidean and Unitary Spaces)

3.1. Euclidean Spaces E[subscript n]

3.2. Unitary Spaces U[subscript n] (or Complex Inner-product Vector Spaces)

3.3. Orthonormal Bases and the Gram-Schmidt Procedure for Orthonormalization of Bases

3.4. Direct and Orthogonal Sums of Subspaces and the Orthogonal Complement of a Subspace

3.4.1. Direct and Orthogonal Sums of Subspaces

3.4.2. The Orthogonal Complement of a Subspace

4. Dual Spaces and the Change of Basis

4.1. The Dual Space U*[subscript n] of a Unitary Space U[subscript n]

4.2. The Adjoint Operator

4.3. The Change of Bases in V[subscript n](F)

4.3.1. The Change of the Matrix-Column [xi] that Represents a Vector x [set membership] V[subscript n](F) (Contravariant Vectors)

4.3.2. The Change of the n x n Matrix A That Represents an Operator A [set membership] L(V[subscript n](F), V[subscript n](F)) (Mixed Tensor of the Second Order)

4.4. The Change of Bases in Euclidean (E[subscript n]) and Unitary (U[subscript n]) Vector Spaces

4.5. The Change of Biorthogonal Bases in V*[subscript n](F) (Covariant Vectors)

4.6. The Relation between V[subscript n](F) and V*[subscript n](F) is Symmetric (The Invariant Isomorphism between V[subscript n](F) and V**[subscript n] (F))

4.7. Isodualism-The Invariant Isomorphism between the Superspaces L(V[subscript n](F), V[subscript n](F)) and L(V*[subscript n](F),V*[subscript n](F))

5. The Eigen Problem or Diagonal Form of Representing Matrices

5.1. Eigenvalues, Eigenvectors, and Eigenspaces

5.2. Diagonalization of Square Matrices

5.3. Diagonalization of an Operator in U[subscript n]

5.3.1. Two Examples of Normal Matrices

5.4. The Actual Method for Diagonalization of a Normal Operator

5.5. The Most Important Subsets of Normal Operators in U[subscript n]

5.5.1. The Unitary Operators A[superscript dagger] = A[superscript -1]

5.5.2. The Hermitian Operators A[superscript dagger] = A

5.5.3. The Projection Operators P[superscript dagger] = P = P[superscript 2]

5.5.4. Operations with Projection Operators

5.5.5. The Spectral Form of a Normal Operator A

5.6. Diagonalization of a Symmetric Operator in E[subscript 3]

5.6.1. The Actual Procedure for Orthogonal Diagonalization of a Symmetric Operator in E[subscript 3]

5.6.2. Diagonalization of Quadratic Forms

5.6.3. Conic Sections in R[superscript 2]

5.7. Canonical Form of Orthogonal Matrices

5.7.1. Orthogonal Matrices in R[superscript n]

5.7.2. Orthogonal Matrices in R[superscript 2] (Rotations and Reflections)

5.7.3. The Canonical Forms of Orthogonal Matrices in R[superscript 3] (Rotations and Rotations with Inversions)

6. Tensor Product of Unitary Spaces

6.1. Kronecker Product of Matrices

6.2. Axioms for the Tensor Product of Unitary Spaces

6.2.1. The Tensor product of Unitary Spaces C[superscript m] and C[superscript n]

6.2.2. Definition of the Tensor Product of Unitary Spaces, in Analogy with the Previous Example

6.3. Matrix Representation of the Tensor Product of Unitary Spaces

6.4. Multiple Tensor Products of a Unitary Space U[subscript n] and of its Dual Space U*[subscript n] as the Principal Examples of the Notion of Unitary Tensors

6.5. Unitary Space of Antilinear Operators L[subscript a] (U[subscript m], U[subscript n]) as the Main Realization of U[subscript m] [plus sign in circle] U[subscript n]

6.6. Comparative Treatment of Matrix Representations of Linear Operators from L(U[subscript m], U[subscript n]) and Antimatrix Representations of Antilinear Operators from L[subscript a] (U[subscript m], U[subscript n]) = U[subscript m] [plus sign in circle] U[subscript n]

7. The Dirac Notation in Quantum Mechanics: Dualism between Unitary Spaces (Sect. 4.1) and Isodualism between Their Superspaces (Sect. 4.7)

7.1. Repeating the Statements about the Dualism D

7.2. Invariant Linear and Antilinear Bijections between the Superspaces L(U[subscript n], U[subscript n]) and L(U*[subscript n], U*[subscript n])

7.2.1. Dualism between the Superspaces

7.2.2. Isodualism between Unitary Superspaces

7.3. Superspaces L(U[subscript n], U[subscript n]) [characters not reproducible] L(U*[subscript n], U*[subscript n]) as the Tensor Product of U[subscript n] and U*[subscript n], i.e., U[subscript n] [plus sign in circle] U*[subscript n]

7.3.1. The Tensor Product of U[subscript n] and U*[subscript n]

7.3.2. Representation and the Tensor Nature of Diads

7.3.3. The Proof of Tensor Product Properties

7.3.4. Diad Representations of Operators

Bibliography

Index