TY - BOOK AU - Vujičić,Milan TI - Linear algebra thoroughly explained / SN - 9783540746379 (hbk.) U1 - 512.5 22 PY - 2008/// CY - Berlin PB - Springer KW - Algebras, Linear N1 - Includes bibliographical references and index; 1 Vector Spaces 2 1.1 Introduction 3 1.2 Geometrical Vectors in a Plane 4 1.3 Vectors in a Cartesian (Analytic) Plane R2 5 1.4 Scalar Multiplication (The Product of a Number with a Vector) 6 1.5 The Dot Product of Two Vectors (or the Euclidean Inner Product 7 of Two Vectors in R2) 8 1.6 Applications of the Dot Product and Scalar Multiplication 9 1.7 Vectors in Three-Dimensional Space (Spatial Vectors) ............ 15 10 1.8 The Cross Product inR3 .................... ................. 18 11 1.9 The Mixed Triple Product in RR3. Applications of the Cross 12 and M ixed Products ........................... ............ 21 13 1.10 Equations of Lines in Three-Dimensional Space ................. 24 14 1.11 Equations of Planes in Three-Dimensional Space ................ 26 15 1.12 Real Vector Spaces and Subspaces ......................... 28 16 1.13 Linear Dependence and Independence. Spanning Subsets and Bases 30 17 1.14 The Three Most Important Examples of Finite-Dimensional Real 18 Vector Spaces .............................................. 33 19 1.14.1 The Vector Space Rn" (Number Columns) ................ 33 20 1.14.2 The Vector Space Rn,xn (Matrices) ...................... 35 21 1.14.3 The Vector Space P3 (Polynomials) ..................... 37 22 1.15 Some Special Topics about Matrices ......................... 39 23 1.15.1 Matrix Multiplication .............. .............. 39 24 1.15.2 Some Special Matrices ............................ 40 25 A Determinants ............................................ 45 A. I Definitions of Determinants ............... ............... 45 26 A.2 Properties of Determinants ................. ............... 49 27 2 Linear Mappings and Linear Systems ........................ 59 28 2.1 A Short Plan for the First 5 Sections of Chapter 2 ................ 59 29 2.2 Some General Statements about Mapping ...................... 60 30 2.3 The Definition of Linear Mappings (Linmaps) .................. 62 31 2.4 The Kernel and the Range of L ............................. 63 31.1.1.1.1 L' 32 2.5 The Quotient Space V,/ker L and the Isomorphism Vn/ker - ran L 65 33 2.6 Representation Theory ................ ................... 67 34 2.6.1 The Vector Space L(V,,Wm) .......................... 68 35 2.6.2 The Linear Map M:R - R'm ........................ 69 36 2.6.3 The Three Isomorphisms v, w and v - w ................. 70 37 2.6.4 How to Calculate the Representing Matrix M ............. 72 38 2.7 An Example (Representation of a Linmap Which Acts between 39 Vector Spaces of Polynomials) ............................. 75 40 2.8 Systems of Linear Equations (Linear Systems) .................. 79 41 2.9 The Four Tasks ...................................... 85 42 2.10 The Column Space and the Row Space ........................ 86 43 2.11 Two Examples of Linear Dependence of Columns 44 and Rows of a Matrix ...................................... 88 45 2.12 Elementary Row Operations (Eros) and Elementary Matrices ...... 91 46 2.12.1 Eros ........................................ 91 47 2.12.2 Elementary Matrices ............................. 93 48 2.13 The GJ Form of a Matrix ........................ ........... 95 49 2.14 An Example (Preservation of Linear Independence 50 and Dependence in GJ Form) ................................. 97 51 2.15 The Existence of the Reduced Row-Echelon (GJ) 52 Form for Every Matrix ................................. 99 53 2.16 The Standard Method for Solving A = b ..................... 101 54 2.16.1 When Does a Consistent System A-f = b Have 54.1 a Unique Solution? ........... ...................... 102 55 2.16.2 When a Consistent System A- = b Has No 55.1 Unique Solution ..................................... 108 56 2.17 The GJM Procedure - a New Approach to Solving Linear Systems 57 with Nonunique Solutions ................ ................ 109 58 2.17.1 Detailed Explanation ................................ 110 59 2.18 Summary of Methods for Solving Systems of Linear Equations .... 116 60 3 Inner-Product Vector Spaces (Euclidean and Unitary Spaces) ....... 119 61 3.1 Euclidean Spaces E, ........................................ 119 62 3.2 Unitary Spaces U, (or Complex Inner-product Vector Spaces) ..... 126 63 3.3 Orthonormal Bases and the Gram-Schmidt Procedure 64 for Orthonormalization of Bases .............................. 131 65 3.4 Direct and Orthogonal Sums of Subspaces and the Orthogonal 66 Complement of a Subspace ................... ............. 139 67 3.4.1 Direct and Orthogonal Sums of Subspaces ............... 139 68 3.4.2 The Orthogonal Complement of a Subspace .............. 141 69 4 Dual Spaces and the Change of Basis ........................ 145 70 4.1 The Dual Space U,* of a Unitary Space U, .................... 145 71 4.2 The Adjoint Operator ................ ................... . 153 72 4.3 The Change of Bases in V,(F) ................................ 157 73 4.3.1 The Change of the Matrix-Column 4 That Represents 73.1 a Vector x E V,(F) (Contravariant Vectors) ............... 158 74 4.3.2 The Change of the n x n Matrix a.d That Represents 74.1 an Operator A E L(V,,(F),V,n(F)) (Mixed Tensor 74.2 of the Second Order) ................................. 159 75 4.4 The Change of Bases in Euclidean (E,) and Unitary (U,) Vector 76 Spaces........................ . .... ...... ... . ............ 162 77 4.5 The Change of Biorthogonal Bases in V*(F) 78 (Covariant Vectors) ......................................... 164 79 4.6 The Relation between V,(F) and V(F) is Symmetric 80 (The Invariant Isomorphism between V,n(F) and V* (F)) .......... 167 81 4.7 Isodualism-The Invariant Isomorphism between the Superspaces 82 L(V,(F),V,,(F)) andL(Vn,(F), V* (F)) ......................... 168 83 5 The Eigen Problem or Diagonal Form of Representing Matrices ..... 173 84 5.1 Eigenvalues, Eigenvectors, and Eigenspaces .................... 173 85 5.2 Diagonalization of Square Matrices ......................... 180 86 5.3 Diagonalization of an Operator in U, ......................... 183 87 5.3.1 Two Examples of Normal Matrices .................. .. 188 88 5.4 The Actual Method for Diagonalization of a Normal Operator ..... 191 89 5.5 The Most Important Subsets of Normal Operators in U, .......... 194 90 5.5.1 The Unitary Operators At = A- ..................... 194 91 5.5.2 The Hermitian Operators At = A ....................... 198 92 5.5.3 The Projection Operators Pt = p = P2 .................. 200 93 5.5.4 Operations with Projection Operators ................... 203 94 5.5.5 The Spectral Form of a Normal Operator A ............... 207 95 5.6 Diagonalization of a Symmetric Operator in E3 ................. 208 96 5.6.1 The Actual Procedure for Orthogonal Diagonalization 96.1 of a Symmetric Operator in E3 ......................... 214 97 5.6.2 Diagonalization of Quadratic Forms .................... 218 98 5.6.3 Conic Sections in 1R2 .............................. 220 99 5.7 Canonical Form of Orthogonal Matrices ....................... 228 100 5.7.1 Orthogonal Matrices in 1R ......... ............... 228 101 5.7.2 Orthogonal Matrices in R2 (Rotations and Reflections) ..... 229 102 5.7.3 The Canonical Forms of Orthogonal Matrices in R3 102.1 (Rotations and Rotations with Inversions) ................ 240 103 6 Tensor Product of Unitary Spaces ........................... 243 104 6.1 Kronecker Product of Matrices ........................... . 243 105 6.2 Axioms for the Tensor Product of Unitary Spaces ................ 247 106 6.2.1 The Tensor product of Unitary Spaces Cm and Cn ......... 247 107 6.2.2 Definition of the Tensor Product of Unitary Spaces, 107.1 in Analogy with the Previous Example ............ . 249 108 6.3 Matrix Representation of the Tensor Product of Unitary Spaces .... 250 109 6.4 Multiple Tensor Products of a Unitary Space U, and of its Dual 110 Space U,* as the Principal Examples of the Notion of Unitary 111 Tensors ..................................................252 112 6.5 Unitary Space of Antilinear Operators La(U, Un) as the Main 113 Realization of Urn Un ....................... .............. 254 114 6.6 Comparative Treatment of Matrix Representations of Linear 115 Operators from L(Um, U,,) and Antimatrix Representations 116 of Antilinear Operators from La (U., Un,) = U,rn Un ............. 257 117 7 The Dirac Notation in Quantum Mechanics: Dualism 118 between Unitary Spaces (Sect. 4.1) and Isodualism 119 between Their Superspaces (Sect. 4.7) .......................... 263 120 7.1 Repeating the Statements about the Dualism D .................. 263 121 7.2 Invariant Linear and Antilinear Bijections between 122 the Superspaces L(U,,, U,) and L(U,, U,*) 123 7.2.1 Dualism between the Superspaces 124 7.2.2 Isodualism between Unitary Superspaces 125 7.3 Superspaces L(U,, Un) c L(Ul, U*) as the Tensor Product of U, 126 and Ut , i.e., U 127 7.3.1 The Tensor Product of U, and UW 128 7.3.2 Representation and the Tensor Nature of Diads 129 7.3.3 The Proof of Tensor Product Properties 130 7.3.4 Diad Representations of Operators 131 Bibliography ER -