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1 Vector Spaces <br/>2 1.1 Introduction <br/>3 1.2 Geometrical Vectors in a Plane <br/>4 1.3 Vectors in a Cartesian (Analytic) Plane R2 <br/>5 1.4 Scalar Multiplication (The Product of a Number with a Vector) <br/>6 1.5 The Dot Product of Two Vectors (or the Euclidean Inner Product<br/>7 of Two Vectors in R2) <br/>8 1.6 Applications of the Dot Product and Scalar Multiplication <br/>9 1.7 Vectors in Three-Dimensional Space (Spatial Vectors) ............ 15<br/>10 1.8 The Cross Product inR3 .................... ................. 18<br/>11 1.9 The Mixed Triple Product in RR3. Applications of the Cross<br/>12 and M ixed Products ........................... ............ 21<br/>13 1.10 Equations of Lines in Three-Dimensional Space ................. 24<br/>14 1.11 Equations of Planes in Three-Dimensional Space ................ 26<br/>15 1.12 Real Vector Spaces and Subspaces ......................... 28<br/>16 1.13 Linear Dependence and Independence. Spanning Subsets and Bases 30<br/>17 1.14 The Three Most Important Examples of Finite-Dimensional Real<br/>18 Vector Spaces .............................................. 33<br/>19 1.14.1 The Vector Space Rn" (Number Columns) ................ 33<br/>20 1.14.2 The Vector Space Rn,xn (Matrices) ...................... 35<br/>21 1.14.3 The Vector Space P3 (Polynomials) ..................... 37<br/>22 1.15 Some Special Topics about Matrices ......................... 39<br/>23 1.15.1 Matrix Multiplication .............. .............. 39<br/>24 1.15.2 Some Special Matrices ............................ 40<br/>25 A Determinants ............................................ 45<br/>A. I Definitions of Determinants ............... ............... 45<br/>26 A.2 Properties of Determinants ................. ............... 49<br/>27 2 Linear Mappings and Linear Systems ........................ 59<br/>28 2.1 A Short Plan for the First 5 Sections of Chapter 2 ................ 59<br/>29 2.2 Some General Statements about Mapping ...................... 60<br/>30 2.3 The Definition of Linear Mappings (Linmaps) .................. 62<br/>31 2.4 The Kernel and the Range of L ............................. 63<br/>31.1.1.1.1 L'<br/>32 2.5 The Quotient Space V,/ker L and the Isomorphism Vn/ker - ran L 65<br/>33 2.6 Representation Theory ................ ................... 67<br/>34 2.6.1 The Vector Space L(V,,Wm) .......................... 68<br/>35 2.6.2 The Linear Map M:R - R'm ........................ 69<br/>36 2.6.3 The Three Isomorphisms v, w and v - w ................. 70<br/>37 2.6.4 How to Calculate the Representing Matrix M ............. 72<br/>38 2.7 An Example (Representation of a Linmap Which Acts between<br/>39 Vector Spaces of Polynomials) ............................. 75<br/>40 2.8 Systems of Linear Equations (Linear Systems) .................. 79<br/>41 2.9 The Four Tasks ...................................... 85<br/>42 2.10 The Column Space and the Row Space ........................ 86<br/>43 2.11 Two Examples of Linear Dependence of Columns<br/>44 and Rows of a Matrix ...................................... 88<br/>45 2.12 Elementary Row Operations (Eros) and Elementary Matrices ...... 91<br/>46 2.12.1 Eros ........................................ 91<br/>47 2.12.2 Elementary Matrices ............................. 93<br/>48 2.13 The GJ Form of a Matrix ........................ ........... 95<br/>49 2.14 An Example (Preservation of Linear Independence<br/>50 and Dependence in GJ Form) ................................. 97<br/>51 2.15 The Existence of the Reduced Row-Echelon (GJ)<br/>52 Form for Every Matrix ................................. 99<br/>53 2.16 The Standard Method for Solving A = b ..................... 101<br/>54 2.16.1 When Does a Consistent System A-f = b Have<br/>54.1 a Unique Solution? ........... ...................... 102<br/>55 2.16.2 When a Consistent System A- = b Has No<br/>55.1 Unique Solution ..................................... 108<br/>56 2.17 The GJM Procedure - a New Approach to Solving Linear Systems<br/>57 with Nonunique Solutions ................ ................ 109<br/>58 2.17.1 Detailed Explanation ................................ 110<br/>59 2.18 Summary of Methods for Solving Systems of Linear Equations .... 116<br/>60 3 Inner-Product Vector Spaces (Euclidean and Unitary Spaces) ....... 119<br/>61 3.1 Euclidean Spaces E, ........................................ 119<br/>62 3.2 Unitary Spaces U, (or Complex Inner-product Vector Spaces) ..... 126<br/>63 3.3 Orthonormal Bases and the Gram-Schmidt Procedure<br/>64 for Orthonormalization of Bases .............................. 131<br/>65 3.4 Direct and Orthogonal Sums of Subspaces and the Orthogonal<br/>66 Complement of a Subspace ................... ............. 139<br/>67 3.4.1 Direct and Orthogonal Sums of Subspaces ............... 139<br/>68 3.4.2 The Orthogonal Complement of a Subspace .............. 141<br/>69 4 Dual Spaces and the Change of Basis ........................ 145<br/>70 4.1 The Dual Space U,* of a Unitary Space U, .................... 145<br/>71 4.2 The Adjoint Operator ................ ................... . 153<br/>72 4.3 The Change of Bases in V,(F) ................................ 157<br/>73 4.3.1 The Change of the Matrix-Column 4 That Represents<br/>73.1 a Vector x E V,(F) (Contravariant Vectors) ............... 158<br/>74 4.3.2 The Change of the n x n Matrix a.d That Represents<br/>74.1 an Operator A E L(V,,(F),V,n(F)) (Mixed Tensor<br/>74.2 of the Second Order) ................................. 159<br/>75 4.4 The Change of Bases in Euclidean (E,) and Unitary (U,) Vector<br/>76 Spaces........................ . .... ...... ... . ............ 162<br/>77 4.5 The Change of Biorthogonal Bases in V*(F)<br/>78 (Covariant Vectors) ......................................... 164<br/>79 4.6 The Relation between V,(F) and V(F) is Symmetric<br/>80 (The Invariant Isomorphism between V,n(F) and V* (F)) .......... 167<br/>81 4.7 Isodualism-The Invariant Isomorphism between the Superspaces<br/>82 L(V,(F),V,,(F)) andL(Vn,(F), V* (F)) ......................... 168<br/>83 5 The Eigen Problem or Diagonal Form of Representing Matrices ..... 173<br/>84 5.1 Eigenvalues, Eigenvectors, and Eigenspaces .................... 173<br/>85 5.2 Diagonalization of Square Matrices ......................... 180<br/>86 5.3 Diagonalization of an Operator in U, ......................... 183<br/>87 5.3.1 Two Examples of Normal Matrices .................. .. 188<br/>88 5.4 The Actual Method for Diagonalization of a Normal Operator ..... 191<br/>89 5.5 The Most Important Subsets of Normal Operators in U, .......... 194<br/>90 5.5.1 The Unitary Operators At = A- ..................... 194<br/>91 5.5.2 The Hermitian Operators At = A ....................... 198<br/>92 5.5.3 The Projection Operators Pt = p = P2 .................. 200<br/>93 5.5.4 Operations with Projection Operators ................... 203<br/>94 5.5.5 The Spectral Form of a Normal Operator A ............... 207<br/>95 5.6 Diagonalization of a Symmetric Operator in E3 ................. 208<br/>96 5.6.1 The Actual Procedure for Orthogonal Diagonalization<br/>96.1 of a Symmetric Operator in E3 ......................... 214<br/>97 5.6.2 Diagonalization of Quadratic Forms .................... 218<br/>98 5.6.3 Conic Sections in 1R2 .............................. 220<br/>99 5.7 Canonical Form of Orthogonal Matrices ....................... 228<br/>100 5.7.1 Orthogonal Matrices in 1R ......... ............... 228<br/>101 5.7.2 Orthogonal Matrices in R2 (Rotations and Reflections) ..... 229<br/>102 5.7.3 The Canonical Forms of Orthogonal Matrices in R3<br/>102.1 (Rotations and Rotations with Inversions) ................ 240<br/>103 6 Tensor Product of Unitary Spaces ........................... 243<br/>104 6.1 Kronecker Product of Matrices ........................... . 243<br/>105 6.2 Axioms for the Tensor Product of Unitary Spaces ................ 247<br/>106 6.2.1 The Tensor product of Unitary Spaces Cm and Cn ......... 247<br/>107 6.2.2 Definition of the Tensor Product of Unitary Spaces,<br/>107.1 in Analogy with the Previous Example ............ . 249<br/>108 6.3 Matrix Representation of the Tensor Product of Unitary Spaces .... 250<br/>109 6.4 Multiple Tensor Products of a Unitary Space U, and of its Dual<br/>110 Space U,* as the Principal Examples of the Notion of Unitary<br/>111 Tensors ..................................................252<br/>112 6.5 Unitary Space of Antilinear Operators La(U, Un) as the Main<br/>113 Realization of Urn Un ....................... .............. 254<br/>114 6.6 Comparative Treatment of Matrix Representations of Linear<br/>115 Operators from L(Um, U,,) and Antimatrix Representations<br/>116 of Antilinear Operators from La (U., Un,) = U,rn Un ............. 257<br/>117 7 The Dirac Notation in Quantum Mechanics: Dualism<br/>118 between Unitary Spaces (Sect. 4.1) and Isodualism<br/>119 between Their Superspaces (Sect. 4.7) .......................... 263<br/>120 7.1 Repeating the Statements about the Dualism D .................. 263<br/>121 7.2 Invariant Linear and Antilinear Bijections between<br/>122 the Superspaces L(U,,, U,) and L(U,, U,*)<br/>123 7.2.1 Dualism between the Superspaces<br/>124 7.2.2 Isodualism between Unitary Superspaces <br/>125 7.3 Superspaces L(U,, Un) c L(Ul, U*) as the Tensor Product of U,<br/>126 and Ut , i.e., U <br/>127 7.3.1 The Tensor Product of U, and UW <br/>128 7.3.2 Representation and the Tensor Nature of Diads <br/>129 7.3.3 The Proof of Tensor Product Properties <br/>130 7.3.4 Diad Representations of Operators <br/>131 Bibliography<br/><br/> |
