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A
Absolute value (complex), 12
Abelian group, 405
Adjoint, classical, 273
Algebraic multiplicity, 300
Alternating mappings, 278, 362, 401
Associate, 408
Associated homogeneous system, 83
Augmented matrix, 59
B
orthogonal, 245
orthonormal, 245
second dual, 369
standard, 125
usual, 125
Basis-finding algorithm, 127
Bessel inequality, 254
Bijective mapping, 168
alternating, 278
matrix representation of, 362
polar form of, 365
real symmetric, 365
symmetric, 363
determinants, 419
Jordan, 345
square, 40
Bounded, 158
C
Cancellation law, 113
rational, 333
row, 74
triangular, 327
Casting-out algorithm, 128
Cauchy–Schwarz inequality, 5, 231, 242
Cayley–Hamilton theorem, 296, 310
Change-of-basis (transition) matrix, 201
Change-of-coordinate matrix, 223
Characteristic polynomial, 296, 307
value, 298
Classical adjoint, 273
matrix, 59
Cofactor, 271
Column, 27
matrix, 27
operations, 89
space, 120
vector, 3
Colsp(A), column space, 126
Commutative law, 405
group, 113
Commuting (diagram), 398
Companion matrix, 306
Complement, orthogonal, 244
Complementary minor, 275
Completing the square, 395
Complex:
conjugate, 13
inner product, 241
n-space, 13
plane, 12
Complexity, 88
Components, 2
Composition of mappings, 167
Congruent matrices, 362
diagonalization, 61
Conjugate:
complex, 12
linearity, 241
matrix, 38
symmetric, 241
Consistent system, 59
Convex set, 195
vector, 130
Cramer’s rule, 274
Cross product, 10
Curves, 8
group, 410
D
Δij, Kronecker delta function, 33
Decomposable, 329
Decomposition:
direct-sum, 129
primary, 240
Degenerate, 362
bilinear form, 362
linear equations, 59
Dependence, linear, 133
Derivative, 170
linear operator, 277
Diagonal, 32
blocks, 40
quadratic form, 304
Diagonal (of a matrix), 10
Diagonalization:
algorithm, 301
in inner product space, 384
Dimension of solution spaces, 82
Dimension of vector spaces, 82, 141
finite, 124
infinite, 124
subspaces, 126
decomposition, 329
Directed line segment, 7
Divides, 414
Division algorithm, 414
Dot product, 4
Dual:
E
Echelon:
matrices, 70
Eigenline, 298
Eigenspace, 301
Elementary divisors, 333
Elementary matrix, 84
Elementary operations, 61
column, 86
Elimination, Gaussian, 67
Empty set, θ, 112
Equal:
functions, 166
matrices, 27
vectors, 2
Equations (See Linear equations)
Equivalence:
classes, 418
matrix, 87
row, 72
Equivalent systems, 61
Exterior product, 403
F
Finite dimension, 124
Form:
bilinear, 361
linear, 351
quadratic, 365
Forward elimination 63, 67, 73
series, 235
Full rank, 41
factorization, 419
Function, 156
space F(X ), 114
Functional, linear, 351
Fundamental Theorem of Algebra, 416
G
Gaussian elimination, 61, 67, 73
Gaussian integer, 411
Gauss–Jordan algorithm, 74
General solution, 58
Geometric multiplicity, 300
Gram–Schmidt orthogonalization, 237
Graph, 166
Greatest common divisor, 415
H
Hermitian:
form, 366
quadratic form, 366
Hilbert space, 331
Hom(V, U ), 175
I
i, imaginary, 12
Ideal, 407
matrix, 33
ijk notation, 9
Imaginary part, 12
Im F, image, 171
Im z, imaginary part, 12
Inclusion mapping, 192
Inconsistent systems, 59
Independence, linear, 133
Index, 30
Index of nilpotency, 330
Inertia, Law of, 366
Infinite dimension, 124
Infinity-norm, 243
Injective mapping, 168
Inner product, 4
complex, 241
Inner product spaces, 228
linear operators on, 379
Integral, 170
domain, 408
Invariance, 226
Invariant subspaces, 226, 328, 334
direct-sum, 329
Inverse image, 166
Inverse mapping, 166
computing, 85
inversion, 269
Invertible:
Irreducible, 408
Isometry, 383
Isomorphic vector spaces, 171, 406
J
Jordan:
block, 306
K
Ker F, kernel, 171
Kronecker delta function δij, 33
L
l2-space, 231
Laplace expansion, 272
Law of inertia, 365
Leading:
coefficient, 60
nonzero element, 70
unknown, 60
Least square solution, 135
Legendre polynomial, 239
Limits (summation), 30
Linear:
combination, 3, 29, 60, 79, 115
dependence, 121
form, 351
functional, 351
independence, 121
span, 119
Linear equation, 57
Linear equations (system), 58
consistent, 59
echelon form, 65
triangular form, 64
Linear mapping (function), 166, 169
kernel, 171
nullity, 173
rank, 173
Linear operator:
adjoint, 379
characteristic polynomial, 306
determinant, 277
on inner product spaces, 379
invertible, 177
matrix representation, 197
Linear transformation (See linear mappings), 169
Located vectors, 7
M
Mm,n, matrix vector space, 114
Mappings (maps), 166
composition of, 167
linear, 169
matrix, 170
Matrices:
congruent, 362
equivalent, 87
similar, 205
Matrix, 27
augmented, 59
change-of-basis, 201
coefficient, 59
companion, 306
diagonal, 35
elementary, 84
equivalence, 87
identity, 33
invertible, 34
nonsingular, 34
normal, 38
orthogonal, 239
positive definite, 240
space, Mm,n, 114
square root, 298
triangular, 36
Matrix mapping, 167
Matrix multiplication, 30
Matrix representation, 197, 240, 362
bilinear form, 361
change of basis, 201
linear mapping, 197
Metric space, 243
Minkowski’s inequality, 5
principle, 275
Module, 409
Moore–Penrose inverse, 134
Multiplicity, 300
N
n-linear, 278
n-space, 2
complex, 13
real, 2
Natural mapping, 353
New basis, 201
Nonnegative semideflnite, 228
Nonsingular, 112
linear maps, 174
matrices, 34
Normal, 7
matrix, 38
Normalized, 229
Normed vector space, 243
Nullity, 173
nullsp(A), 172
Null space, 172
O
Old basis, 201
One-norm, 243
One-to-one:
correspondence, 168
mapping, 168
Onto mapping (function), 168
Operators (See Linear operators)
Order, n:
determinant, 266
of a group, 410
basis, 233
complement, 233
matrix, 239
operator, 382
projection, 386
substitution, 304
Orthogonalization, Gram–Schmidt, 237
Orthogonally equivalent, 382
Orthonormal, 235
Outer product, 10
P
Parameter, 64
form, 65
Particular solution, 58
Partition, 418
Perpendicular, 4
row reduction, 94
variables, 65
Pivoting (row reduction), 94
Polar form, 365
Polynomial, 413
minimum, 305
space, Pn(t), 114
Positive definite, 228
Positive operators, 228
square root, 393
Primary decomposition theorem, 330
Prime ideal, 412
Principle ideal ring, 408
Principle minor, 275
Product:
exterior, 403
inner, 4
tensor, 398
Projections, 169, 236, 346, 386
orthogonal, 386
Proper value, 298
vector, 298
Pythagorean theorem, 235
Q
Q, rational numbers, 11
Quotient
group, 405
ring, 408
R
Rn, real n-space, 2
Rational:
canonical form, 333
numbers, Q, 11
Real:
numbers, R, 1
part (complex number), 12
Real symmetric bilinear form, 365
Reduce, 73
Relation, 417
Representatives, 418
Restriction mapping, 194
Right-handed system, 11
Right inverse, 191
Ring, 407
quotient, 408
Root, 295
Rotation, 171
Row, 27
canonical form, 72
equivalence, 72
operations, 72
rank, 72
reduce, 73
reduced echelon form, 73
space, 120
S
matrix, 33
multiplication, 33
product, 27
Scaling factor, 298
Schwarz inequality, 5, 231, 242
(See Cauchy–Schwarz inequality)
Second dual space, 352
Self-adjoint operator, 382
Sign of permutation, 269
Signature, 366
Similarity transformation, 205
Singular, 173
Size (matrix), 27
Skew-adjoint operator, 382
Skew-Hermitian, 38
Skew-symmetric, 362
Solution, (linear equations), 57
zero, 121
Spatial vectors, 9
Span, 115
Spanning sets, 115
Spectral theorem, 385
Square:
system of linear equations, 58, 72
Square root of a matrix, 393
Standard:
basis, 125
inner product, 230
Subdiagonal, 306
Subgroup, 405
Subset, 112
Sum of vector spaces, 131
Summation symbol, 29
Superdiagonal, 306
Surjective map, 168
Sylvester’s theorem, 366
Symmetric:
bilinear form, 363
Systems of linear equations, 58
T
Tangent vector, T(t), 9
Target set, 166
Tensor product, 398
Time complexity, 88
Top-down, 73
Trace, 33
Transformation (linear), 169
Transition matrix, 201
Transpose:
linear functional (dual space), 353
matrix, 32
Triangle inequality, 232
Triangular form, 64
block, 40
Triple product, 11
Two-norm, 243
U
Unique factorization domain, 408, 416
matrix, 33
Universal mapping principle (UMP), 398
Usual:
basis, 125
inner product, 230
V
Vandermonde determinant, 292
Variable, free, 65
Vector, 2
coordinates, 130
located, 7
product, 10
spatial, 9
basis, 124
dimension, 124
Volume, 276
W
Wedge (exterior) product, 403
Z
Z, integers, 408
Zero:
matrix, 27
polynomial, 413
solution, 121
vector, 2