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Finkbeiner D.T. Ч Introduction to Matrices and Linear Transformations
Finkbeiner D.T. Ч Introduction to Matrices and Linear Transformations

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Ќазвание: Introduction to Matrices and Linear Transformations

јвтор: Finkbeiner D.T.

јннотаци€:

The present text is a revised form of mimeographed lecture notes written in 1951 for the second semester of a survey course in abstract algebra at Kenyon College.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1960

 оличество страниц: 255

ƒобавлена в каталог: 14.02.2013

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$n$-tuples, ordered      209
$n$-tuples, space of      27 35Ч36 42
$n^{th}$ roots of unity      20 124 140
Abstract systems      1Ч22 211Ч212
Abstract systems, homomorphism of      212Ч213 216Ч221
Abstract systems, isomorphism of      212Ч213 217Ч221
Adjoint      97
Adjoint, determinant of      100
Adjoint, method for $A^{-1}$      97Ч99
Alias interpretation      60
Alibi interpretation      60
angle      162 173Ч181
Annihilator      76
Associativity      7 16Ч18 49 78
Basis      35Ч39 58Ч59
Basis, change of      121
Basis, dual      76
Basis, for idempotent transformations      62
Basis, for nilpotent transformations      148Ч150
Basis, normal orthogonal      176Ч180 186 189
Basis, of null space      53
Basis, orthogonal      176
BesselТs inequality      178
Bilinear form      163Ч168
Bilinear function      162Ч168
Bilinear function, conjugate      170
Bilinear function, rank of      164
Bilinearity      49 78 170
Binary operation      14 16 211
Binary relation      209Ч210
Binomial coefficient      197
Block multiplication      80Ч83
Boolean algebra      15
Canonical form      107
Canonical form, for congruence      166Ч 167 187
Canonical form, for equivalence      118 120
Canonical form, for idempotent matrices      125
Canonical form, for nilpotent matrices      152
Canonical form, for row equivalence      115
Canonical form, for similarity      156
Canonical form, for skew-symmetric matrices      166
Canonical form, for symmetric matrices      166Ч167 187
Canonical form, Jordan      153Ч158
Cartesian product      8 162 209
Cauchy inequality      172
Cayley Ч Hamilton theorem      see Hamilton Ч Cayley theorem
CayleyТs Theorem      219
Characteristic, equation      128 131 133 135 206
Characteristic, numbers      127
Characteristic, polynomial      128 133 136 157Ч159 185
Characteristic, roots      127
Characteristic, values      126Ч 133 136Ч141 146 155 185Ч191 198Ч202 206-207
Characteristic, vectors      126Ч133 135Ч141 185Ч 186 189 191 201-202
Closed operations      13 17Ч19 49
Cofactor      95
Column index      64
Column vector      71 77 89 177
Combinatorial equivalence      104 222Ч228
Commutative group      19
Commutativity      7 16Ч19 48 68
Complement of a set      6
Complex $n^{th}$ roots of unity      20 124 140
complex numbers      169Ч172 189Ч191
Congruence of integers      219Ч220
Congruence of matrices      165Ч168 180 186Ч 187
Conjugate, -bilinear      170
Conjugate, of a complex matrix      184Ч185 189
Conjugate, of a complex number      169Ч172
Conjugate, symmetry      170
Conjunctivity, of matrices      166 190Ч191
Consistency condition      87
Continuity of a matrix      202
Convergence      193Ч201
Coordinate system      35Ч36 (see also Basis)
Coset, of a function      215 220Ч221
Coset, of a relation      214 220
Cosets, product of      216
CramerТs Rule      98Ч99
Cyclic subspace      145
Dantzig, G. B.      222
Derivative of a matrix      202Ч204
Determinant      88Ч105 127 204
Determinant, cofactor of a      95
Diagonability of matrices      130Ч133 136Ч141 189
Diagonability of quadratic forms      189
Diagonal matrix      69 72 129Ч132 136Ч142 166Ч167 186 189 200
Difference of sets      9
Difference of sets, symmetric      9Ч10
Difference system      217
Differential equation, infinite      36
Differential equation, matric form of      203 206
Differential equation, of a linear algebra      49
Differential equation, of a vector space      38Ч40
Differential equation, of subspaces      39Ч40
Differential equation, solution space of      28 205 207
Dimension, finite      36Ч37
Direct sum      143Ч155
Direction numbers      175
Disjoint sets      6
Distance      162 172Ч173 176Ч179
Distributivity      7 16
Division algebra      51
Division ring      27
Domain of a function      10 210
Domain of a relation      210
Dot product      165 169 176
Dual, basis      76
Dual, operations      7
Dual, space      75Ч76 162
Echelon form      111
Echelon form, reduced      112 226
Eigenvalues      127
Eigenvectors      127
Element      4 13
Element, idempotent      16Ч17
Element, identity      7 16Ч18 219
Element, inverse      16Ч18 219
Elementary divisors      157Ч158
Elementary matrices      108Ч110 223Ч228
Elementary operations on matrices      108Ч 110 117 223Ч228
Elementary operations on systems of equations      101
Equivalence classes      107 213 215 220 224
Equivalence relations      13 104 106Ч125 164Ч168 190Ч191 213Ч215 220 222-228
Equivalence relations, combinatorial equivalence      104 222Ч228
Equivalence relations, congruence      165Ч168 180
Equivalence relations, conjunctivity      166 190Ч191
Equivalence relations, equivalence      117Ч120 122 164Ч165
Equivalence relations, row equivalence      110Ч116
Equivalence relations, similarity      120Ч126 130 136Ч141 151 156 180 186-187
Equivalent matrices      117Ч120 122 164
Equivalent systems of equations      101 116 227
Euclidean group      179
Euclidean space      170 173 175Ч179
Factor system      217
Field      19Ч22 26
Field, as a vector space      28
Field, finite      20 70 166
Field, of scalar matrices      67 72
Finite field      20 70 166
Finite geometry      3
Finite-dimensional      36Ч37
Fixed point      127
Form, bilinear      163Ч168
Form, Hermitian      190Ч 191
Form, quadratic      182Ч191
Forsythe, G. E.      100
Frobenius      51
Function      10Ч13 210Ч211 220-221
Function, coset of a      215Ч216 220Ч221
Function, domain of a      10 210
Function, of a function      12 45
Function, range of a      10 210
Function, value of a      10
Functions, bilinear      162Ч168
Functions, hermitian      190
Functions, matrices of      202Ч204
Functions, of matrices      195Ч199
Functions, product of      12Ч13
Functions, quadratic      182Ч191
Functions, space of continuous      28 170Ч171
Gram Ч Schmidt orthogonalization      174Ч175 178
Group      18Ч21 26 218Ч221
Group, commutative      19 26 218
Group, euclidean      179
Group, full linear      57
Group, Lorentz      80
Group, of triangular matrices      72
HadamardТs inequality      191
Hamilton      51
Hamilton Ч Cayley theorem      135Ч136 139 158Ч161 195
Hermitian, congruence, see Conjunctivity; Hermitian, form      190Ч191
Hermitian, function      190
Hermitian, matrix      190Ч191
Hermitian, skew      191
Hermitian, symmetry      170
Homogeneous system      86
Homomorphism, kernel of a      52 221
Homomorphism, of abstract systems      212Ч 213 217Ч221
Homomorphism, of vector spaces      41Ч49
Idempotent element      16Ч17
Idempotent linear transformation      48 62 75 143Ч146
Idempotent matrix      70 72 75 125 139
Identity, element      7 16Ч18 219
Identity, linear transformation      46
Identity, matrix      68
Identity, right      21
Independence, see Linear independence Independent vectors, maximal set of      34
Inequality, BesselТs      178
Inequality, Cauchy      172
Inequality, HadamardТs      191
Inequality, Schwarz      171Ч173
Inequality, triangle      173
Infinite-dimensional      36 171
Inner product      169Ч182
Inner product, space      170Ч181
Integral of a matrix      202
Intersection of sets      6 15 17
Intersection of subspaces      30Ч32
Invariant spaces      54 144Ч155
Inverse of a matrix, calculation of, by adjoint method      97Ч99
Inverse of a matrix, calculation of, by approximation      200
Inverse of a matrix, calculation of, by Hamilton Ч Cayley theorem      159Ч161
Inverse of a matrix, calculation of, by pivot operations      103Ч104 227
Inverse of a matrix, calculation of, by row and column operations      118Ч120
Inverse of a matrix, calculation of, by row operations      113Ч114
Inverse of a matrix, calculation of, for an orthogonal matrix      180
Inverse, a linear transformation      56
Inverse, element      16Ч18 219
Inverse, of a mapping      11
Inverse, of a matrix      70 72
Inverse, right      21
Isomorphism, of abstract systems      212Ч213 217Ч221
Isomorphism, of matrices and linear transformations      73Ч76
Isomorphism, of vector spaces      40Ч43
Jordan canonical form      153Ч158 196Ч198 206
Kernel of a homomorphism      52 221
Kronecker delta      68
Latent roots      127
Law of Cosines      177
Left hand notation      45 77 85
Length      162 169 172Ч179
Linear equations, equivalence of      101 116 227
Linear equations, homogeneous      86
Linear equations, non-homogeneous      86
Linear equations, operations on      100Ч105
Linear equations, solution by CramerТs rule      98Ч99
Linear equations, solution of      84 222
Linear equations, system of      84Ч89
Linear transformation      44Ч62
Linear transformation, characteristic values of      126Ч133 136Ч141 146 155 185Ч191 198Ч202 206-207
Linear transformation, characteristic vectors of      126Ч133 135Ч141 185Ч186 191 201-202
Linear transformation, idempotent      48 62 75 143Ч 146
Linear transformation, identity      46
Linear transformation, inverse of      56
Linear transformation, Lorentz      79Ч80
Linear transformation, matric representation of      58Ч64 73 77 121Ч124
Linear transformation, negative      46
Linear transformation, nilpotent      48 54 76 133 142Ч155
Linear transformation, non-singular      55Ч 57 153
Linear transformation, null space of      52Ч54 142 153
Linear transformation, nullity of      52Ч54
Linear transformation, orthogonal      178Ч182 185Ч186
Linear transformation, range space of      52Ч54 114 142 153
Linear transformation, rank of      52Ч54
Linear transformation, restricted to a subspace      144 149 153
Linear transformation, Segre characteristic of      157Ч158
Linear transformation, singular      55
Linear transformation, specific form of      57Ч62
Linear transformation, trace of      160Ч161
Linear transformation, unitary      179 181
Linear transformation, zero      46
Linear transformations, equality of      45
Linear transformations, product of      45Ч46
Linear transformations, scalar multiple of      45Ч 46
Linear transformations, sum of      45Ч46
Linear, algebra      30 49Ч51 73Ч75
Linear, combinations      30
Linear, dependence      32 33
Linear, functional      75Ч76 162
Linear, group      57
Linear, independence      32Ч 39 131Ч132 136 141 146
Linear, programming      101 227
Linear, space; see Vector space Linear equations, consistency condition for      87
Lorentz matrices      79Ч80
Mac Duffee, C. C.      68
Main diagonal      69
Mapping      10Ч13 211Ч221
Mapping, composite      12
Mapping, image under a      10Ч11 211
Mapping, inverse      11
mapping, many-to-one      11 211
Mapping, of a subspace      48
Mapping, one-to- one      11 211
Mapping, reversible      11
Mapping, zero      13
Mappings, product of      13
Mappings, successive      11 21 215Ч217
Markov matrix      79 130
Matric polynomial      134
Matrices, block multiplication of      80Ч83
Matrices, combinatorial equivalence of      104 222Ч228
Matrices, congruence of      165Ч168 180 186Ч187
Matrices, conjunctivity of      166 190Ч191
Matrices, elementary      108Ч110 223Ч228
Matrices, equality of      64
Matrices, equivalence of      117Ч120 122 164Ч165
Matrices, functions of      195Ч199
Matrices, of functions      202Ч 204
Matrices, product of      66 78
Matrices, row equivalence of      110Ч116
Matrices, sequences of      193Ч199
Matrices, similarity of      120Ч126 130 136Ч141 151 156 180 186-187
Matrices, sum of      65
Matrix representing a, bilinear function      164
Matrix representing a, linear transformation      58Ч64 73 77 121Ч124
Matrix representing a, system of equations      84
Matrix, adjoint of      97
Matrix, augmented      86
Matrix, characteristic equation of      128 131 133 135 206
Matrix, characteristic polynomial of      128 133 136 157Ч159 185
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