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



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Íàçâàíèå: Introduction to Matrices and Linear Transformations

Àâòîð: Finkbeiner D.T.

Àííîòàöèÿ:

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.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1960

Êîëè÷åñòâî ñòðàíèö: 255

Äîáàâëåíà â êàòàëîã: 14.02.2013

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
$n$-tuples, ordered      209
$n$-tuples, space of      27 35—36 42
$n^{th}$ roots of unity      20 124 140
Abstract systems      1—22 211—212
Abstract systems, homomorphism of      212—213 216—221
Abstract systems, isomorphism of      212—213 217—221
Adjoint      97
Adjoint, determinant of      100
Adjoint, method for $A^{-1}$      97—99
Alias interpretation      60
Alibi interpretation      60
angle      162 173—181
Annihilator      76
Associativity      7 16—18 49 78
Basis      35—39 58—59
Basis, change of      121
Basis, dual      76
Basis, for idempotent transformations      62
Basis, for nilpotent transformations      148—150
Basis, normal orthogonal      176—180 186 189
Basis, of null space      53
Basis, orthogonal      176
Bessel’s inequality      178
Bilinear form      163—168
Bilinear function      162—168
Bilinear function, conjugate      170
Bilinear function, rank of      164
Bilinearity      49 78 170
Binary operation      14 16 211
Binary relation      209—210
Binomial coefficient      197
Block multiplication      80—83
Boolean algebra      15
Canonical form      107
Canonical form, for congruence      166— 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      166—167 187
Canonical form, Jordan      153—158
Cartesian product      8 162 209
Cauchy inequality      172
Cayley — Hamilton theorem      see Hamilton — Cayley theorem
Cayley’s Theorem      219
Characteristic, equation      128 131 133 135 206
Characteristic, numbers      127
Characteristic, polynomial      128 133 136 157—159 185
Characteristic, roots      127
Characteristic, values      126— 133 136—141 146 155 185—191 198—202 206-207
Characteristic, vectors      126—133 135—141 185— 186 189 191 201-202
Closed operations      13 17—19 49
Cofactor      95
Column index      64
Column vector      71 77 89 177
Combinatorial equivalence      104 222—228
Commutative group      19
Commutativity      7 16—19 48 68
Complement of a set      6
Complex $n^{th}$ roots of unity      20 124 140
complex numbers      169—172 189—191
Congruence of integers      219—220
Congruence of matrices      165—168 180 186— 187
Conjugate, -bilinear      170
Conjugate, of a complex matrix      184—185 189
Conjugate, of a complex number      169—172
Conjugate, symmetry      170
Conjunctivity, of matrices      166 190—191
Consistency condition      87
Continuity of a matrix      202
Convergence      193—201
Coordinate system      35—36 (see also Basis)
Coset, of a function      215 220—221
Coset, of a relation      214 220
Cosets, product of      216
Cramer’s Rule      98—99
Cyclic subspace      145
Dantzig, G. B.      222
Derivative of a matrix      202—204
Determinant      88—105 127 204
Determinant, cofactor of a      95
Diagonability of matrices      130—133 136—141 189
Diagonability of quadratic forms      189
Diagonal matrix      69 72 129—132 136—142 166—167 186 189 200
Difference of sets      9
Difference of sets, symmetric      9—10
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      38—40
Differential equation, of subspaces      39—40
Differential equation, solution space of      28 205 207
Dimension, finite      36—37
Direct sum      143—155
Direction numbers      175
Disjoint sets      6
Distance      162 172—173 176—179
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      75—76 162
Echelon form      111
Echelon form, reduced      112 226
Eigenvalues      127
Eigenvectors      127
Element      4 13
Element, idempotent      16—17
Element, identity      7 16—18 219
Element, inverse      16—18 219
Elementary divisors      157—158
Elementary matrices      108—110 223—228
Elementary operations on matrices      108— 110 117 223—228
Elementary operations on systems of equations      101
Equivalence classes      107 213 215 220 224
Equivalence relations      13 104 106—125 164—168 190—191 213—215 220 222-228
Equivalence relations, combinatorial equivalence      104 222—228
Equivalence relations, congruence      165—168 180
Equivalence relations, conjunctivity      166 190—191
Equivalence relations, equivalence      117—120 122 164—165
Equivalence relations, row equivalence      110—116
Equivalence relations, similarity      120—126 130 136—141 151 156 180 186-187
Equivalent matrices      117—120 122 164
Equivalent systems of equations      101 116 227
Euclidean group      179
Euclidean space      170 173 175—179
Factor system      217
Field      19—22 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      36—37
Fixed point      127
Form, bilinear      163—168
Form, Hermitian      190— 191
Form, quadratic      182—191
Forsythe, G. E.      100
Frobenius      51
Function      10—13 210—211 220-221
Function, coset of a      215—216 220—221
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      162—168
Functions, hermitian      190
Functions, matrices of      202—204
Functions, of matrices      195—199
Functions, product of      12—13
Functions, quadratic      182—191
Functions, space of continuous      28 170—171
Gram — Schmidt orthogonalization      174—175 178
Group      18—21 26 218—221
Group, commutative      19 26 218
Group, euclidean      179
Group, full linear      57
Group, Lorentz      80
Group, of triangular matrices      72
Hadamard’s inequality      191
Hamilton      51
Hamilton — Cayley theorem      135—136 139 158—161 195
Hermitian, congruence, see Conjunctivity; Hermitian, form      190—191
Hermitian, function      190
Hermitian, matrix      190—191
Hermitian, skew      191
Hermitian, symmetry      170
Homogeneous system      86
Homomorphism, kernel of a      52 221
Homomorphism, of abstract systems      212— 213 217—221
Homomorphism, of vector spaces      41—49
Idempotent element      16—17
Idempotent linear transformation      48 62 75 143—146
Idempotent matrix      70 72 75 125 139
Identity, element      7 16—18 219
Identity, linear transformation      46
Identity, matrix      68
Identity, right      21
Independence, see Linear independence Independent vectors, maximal set of      34
Inequality, Bessel’s      178
Inequality, Cauchy      172
Inequality, Hadamard’s      191
Inequality, Schwarz      171—173
Inequality, triangle      173
Infinite-dimensional      36 171
Inner product      169—182
Inner product, space      170—181
Integral of a matrix      202
Intersection of sets      6 15 17
Intersection of subspaces      30—32
Invariant spaces      54 144—155
Inverse of a matrix, calculation of, by adjoint method      97—99
Inverse of a matrix, calculation of, by approximation      200
Inverse of a matrix, calculation of, by Hamilton — Cayley theorem      159—161
Inverse of a matrix, calculation of, by pivot operations      103—104 227
Inverse of a matrix, calculation of, by row and column operations      118—120
Inverse of a matrix, calculation of, by row operations      113—114
Inverse of a matrix, calculation of, for an orthogonal matrix      180
Inverse, a linear transformation      56
Inverse, element      16—18 219
Inverse, of a mapping      11
Inverse, of a matrix      70 72
Inverse, right      21
Isomorphism, of abstract systems      212—213 217—221
Isomorphism, of matrices and linear transformations      73—76
Isomorphism, of vector spaces      40—43
Jordan canonical form      153—158 196—198 206
Kernel of a homomorphism      52 221
Kronecker delta      68
Latent roots      127
Law of Cosines      177
Left hand notation      45 77 85
Length      162 169 172—179
Linear equations, equivalence of      101 116 227
Linear equations, homogeneous      86
Linear equations, non-homogeneous      86
Linear equations, operations on      100—105
Linear equations, solution by Cramer’s rule      98—99
Linear equations, solution of      84 222
Linear equations, system of      84—89
Linear transformation      44—62
Linear transformation, characteristic values of      126—133 136—141 146 155 185—191 198—202 206-207
Linear transformation, characteristic vectors of      126—133 135—141 185—186 191 201-202
Linear transformation, idempotent      48 62 75 143— 146
Linear transformation, identity      46
Linear transformation, inverse of      56
Linear transformation, Lorentz      79—80
Linear transformation, matric representation of      58—64 73 77 121—124
Linear transformation, negative      46
Linear transformation, nilpotent      48 54 76 133 142—155
Linear transformation, non-singular      55— 57 153
Linear transformation, null space of      52—54 142 153
Linear transformation, nullity of      52—54
Linear transformation, orthogonal      178—182 185—186
Linear transformation, range space of      52—54 114 142 153
Linear transformation, rank of      52—54
Linear transformation, restricted to a subspace      144 149 153
Linear transformation, Segre characteristic of      157—158
Linear transformation, singular      55
Linear transformation, specific form of      57—62
Linear transformation, trace of      160—161
Linear transformation, unitary      179 181
Linear transformation, zero      46
Linear transformations, equality of      45
Linear transformations, product of      45—46
Linear transformations, scalar multiple of      45— 46
Linear transformations, sum of      45—46
Linear, algebra      30 49—51 73—75
Linear, combinations      30
Linear, dependence      32 33
Linear, functional      75—76 162
Linear, group      57
Linear, independence      32— 39 131—132 136 141 146
Linear, programming      101 227
Linear, space; see Vector space Linear equations, consistency condition for      87
Lorentz matrices      79—80
Mac Duffee, C. C.      68
Main diagonal      69
Mapping      10—13 211—221
Mapping, composite      12
Mapping, image under a      10—11 211
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 21 215—217
Markov matrix      79 130
Matric polynomial      134
Matrices, block multiplication of      80—83
Matrices, combinatorial equivalence of      104 222—228
Matrices, congruence of      165—168 180 186—187
Matrices, conjunctivity of      166 190—191
Matrices, elementary      108—110 223—228
Matrices, equality of      64
Matrices, equivalence of      117—120 122 164—165
Matrices, functions of      195—199
Matrices, of functions      202— 204
Matrices, product of      66 78
Matrices, row equivalence of      110—116
Matrices, sequences of      193—199
Matrices, similarity of      120—126 130 136—141 151 156 180 186-187
Matrices, sum of      65
Matrix representing a, bilinear function      164
Matrix representing a, linear transformation      58—64 73 77 121—124
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      128 133 136 157—159 185
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