A = LDLT, 497
A = LLT, 494
A = PBP−1, similarity, 352
A = PDP−1, diagonalizable, 372
Ak = PDkP−1, powers of a matrix, 376
A = PDQ, SVD-decomposition, 501
A = PQP−1, transition, 354
A = QR, 499
A = UBUH, unitarily similar, 468
A = UDUH, Spectral Theorem: Matrix Version, 466
A = UTUH, Schur decomposition, 502
Abel-Ruffini Impossibility Theorem, 584
Addition of
matrices, 187
vectors, 33
Adjoint matrix, classical, 277
Adjoint operator, 482
Adjugate, 277
Algebraic multiplicity, 373
Algorithms
Cholesky factorization, 497
conjugate gradient, 530
determinant, 246
determinant, without scaling, 249
Gram-Schmidt, modified, 437
Gram-Schmidt, normalized, 435
linearly independent set of vectors, 75
reduced row echelon form, 26
Angles between vectors, 415–416
Applications
balanced weights, 96
bending beam, 50
calculating area of parallelogram, 256
calculating areas and volumes, 253
chemistry, 79
coded messages, 281
collision, 424
data smoothing, 152
demographic problems, 545
diet problems, 205
dynamical systems, 380
economic model, 385
electrical circuits, 101
elementary mechanics, 95
feeding bacteria, 22
interpolation, 231
Leontief open model, 548
linear least-squares problem, 510
linear ordinary differential equations, 50
method of least squares, 127
models in economic theory, 176
network problems, 99
partial-fraction decomposition, 126
planes in , 276
population migration, 545
powers of a matrix, 376
predator-prey simulation, 125
signals in electrical engineering, 165
systems of linear differential equations, 388
traffic flow problems, 99
work and forces, 423
World Wide Web searching, 477
Archimedes, 96
Area parallelogram, 256
Area triangle, 260
Areas, calculating, 253
Arithmetic-geometric mean property, 574
Arithmetic mean, 577
Arithmetic progression, 576, 578
Arithmetic sum, 577
Arrays, 6
Associativity
addition, 91
area parallelogram, 256
area triangle, 260
Law for Matrix Multiplication, 199
matrix multiplication, 188–192, 198–200, 202–208
scalar-vector product, 161
vector addition, 161
Attractor, 384
Average, 106
-coordination, 342
Backward dynamical system, 401
Backward elimination phase, 26, 32
Bacteria nutrition problem, 30
Balanced weights, 96
Ball, glancing, 424
Base of triangle, 255
Basic Linear Algebra
Subprograms (BLAS), 26, 32, 113
Basic properties of vector spaces , 90
Basis
orthonormal basis, 412, 421, 435
standard vectors, 311
Battery, sign for flow, 102
Bending beam, 50
Best approximate solution, 440, 444
Binomial coefficient, , 577
Binomial Theorem, 577
Biological species, 133
Bivariant interpolation, 240
Block matrices, 504
addition, 504
inverting 2 × 2 block matrices, 508
matrix multiplication, 504–506
scalar multiplication, 504
solving 2 × 2 block matrices, 506
Branches, 99
Bunyakovsky, Viktor Yakovlevich, 408
, 184
Calculating area of parallelogram, 256, 260
Calculating areas and volumes, 253
Calculating volume parallelogram, 260
Calculation of A−1, 226, 276–278
Cauchy, Augustin Louis, 408
Cauchy-Schwarz Inequality, 409, 580
Cautions, 49, 199, 224, 225, 229, 305, 308
(See Dangerous Pitfalls.)
Cayley, Arthur, 469
Cayley-Hamilton Theorem, 469–470
Centroid, 106
Changing coordinates, 343
Characteristic equation/values, 368, 369, 377
Chebyshev, Pafnuty Lvovich, 345
Chebyshev polynomials, 336, 345, 349, 359, 361
Cholesky, Andre-Louis, 494
Cholesky decomposition/factorization, 494
algorithm, 497
Circle Theorem, 533
Circuit Problem I, 103
Circuit problems, 101
Circuits, one loop, 102
Circuits, two loops, 103
Classical adjoint matrix, 277
Closed model, 177
Closed system, 545
Closed under scalar multiplication, 290
Closed under vector addition, 290
Closure axiom for addition, 160
Closure axiom for scalar-vector product, 161
Closure under addition, 91
Closure under multiplication by real numbers, 99
Coded messages, 281
Coefficient matrix, 7, 42, 230
Cofactor expansions, 266
Cofactor matrix, 264
Cofactors, 266
expansion, 266
Collision, 424
Column index, 7
Column rank, 332
Columns, 11
Col(AT), 304
m × n matrices, 54
Column vectors, 6, 43, 91, 230, 405
Combination of columns, 190–191
Commutativity of addition, 91
Commutativity of vector addition, 161
Compact summation, 43
Complement, orthogonal, 416–417
Complex arithmetic, 581
Complex conjugate pairs, 380
Complex numbers, 159, 184, 581
Component-by-component, 34, 89
Componentwise, 89
Composition of
linear functions/mapping, 151
linear transformations, 200
Computing inverse, 226
using determinants, 276
Conic section, 87
Conjugate gradient algorithm/method, 530–531
Conjugate transpose, 461
Consumption matrix, 178
Continuous real-valued functions, 166
Contraction, 156
Contradiction, 567
Contrapositive, 565
Convergence properties, 541
Converse, 565
Convex quadratic programming problems, 507
Coordinate systems, 341
changing, 343
isomorphic, 317
Coordination, 342
Cosines, Law, 415
Cramer, Gabriel, 275
Current Law, 101
Curve fitting, 127–129, 444–445
(See Linear least squares.)
Cycle, 384
(See Cautions.)
Data motion, 17
Data smoothing, 152
Decomposition, (See Factorization.)
Demand vector, 549
Demographic problems, 545
Denial of a quantified assertion, 572
Dependent. (See Linearly dependent.)
3 × 3 matrices, 269
algorithm without scaling, 249
cofactors, 264
computing, 246
direct calculation methods, 268–269
finding eigenvalues, 369
linear function, 274
nonzero, 251
product of matrices, 271
Properties I, II, III, 244
replacement, 244
review, 282
scaling, 244
swapping, 245
transpose, 273
triangular matrix, 244
without scaling, 249
zero, 250
Diag(d1,d2, …,dn), 399
Diagonal matrix, 193, 372, 377
Diagonalization, 548
Diagonalizable, 372
involving complex numbers, 378
Diagonalization, 548
Diagonally dominant matrix, 532
Diet problems, 205
Difference, 570
Differentiation operator, 308, 349
Dilation, 147
Dimension(s) of
column space, Dim(Col(A)), 230, 332
linear transformation of subspace, 330, 327
null space, Dim(Null(A)), 330
row space, Dim(Row(A)), 231, 332
subspaces, 329
Direct product, 522
Direct sum, 422
Direct verification, 564
Discrete signal processing, 108
Displacement, 424
Distance, between two points, 131
Distance, from a point to a hyperplane, 450
Distance function, 410
Distinct eigenvalues, 374
Distributive Laws, 91
scalar sum times vector, 161
scalar times vector, 161
Dominant eigenvalue, 554
Dot product, 91, 107, 192, 406
(See Inner product.)
analysis, 384
backward/forward, 401
in , 384
Economic models, 385
Eigenproblem, 368
Eigenspace, 398
Eigensystem, without determinants, 389
Eigensystem(s), 367
characteristic equation/polynomial, 369, 377
complex conjugate pairs, 380
distinct, 374
generalized problem, 399
linear operator, 371
matrix, 368
positive definite matrix, 401, 494
powers of matrix, Ak, 376–377, 469
real symmetric matrix, 463
self-adjoint operator, 462
linear operator, 371
matrix, 368
Electrical circuits/networks, 101
determinants, 270
Elementary mechanics, 95, 102–104, 712
Elementary row operations, 9, 15, 16
Ellipse, 476
Enlargement, 147
Equality matrices, 187
Equilibrium, 96
prices, 387
Equivalence relation, 12, 317, 355
Equivalent form of Ax = b, 43
Equivalent properties
linear dependences, 76
m × n matrices, 66
n × n matrices, 229
Equivalent systems of equations, 10–12
Euclidean norm, 407
Euclidean vector space, 88
n-dimensional, 90
Euler, Leonhard, 583
Euler’s Equation, 583
Even function, 454
Existence of additive inverses, 161
Existence of a zero vector, 161
Existential quantifier, 571
Exotic vector space, 183
Expansion using row i/column j, 265, 266
Expansions, 156
Extraneous solutions, 159
Factorization, 487
LDLT , 497
LU of PA, 492
QR, 499
SVD, 501
xn − yn, 574
Feeding bacteria, 22
Field, 164
Fill-in, 525
Final demand vector, 179
Finite difference approximation, 560
Finite-dimensional, vector spaces, 319–320, 328
Fixed point, 158
Flexibility matrix, 51
Floating-point arithmetic, 87
Floating-point values, 24
Flow Axiom, 99
For all, 571
Force, 424
Forward elimination phase, 26, 32
Forward substitution, 490
Fourier analysis, 417
Free parameter/variable, 46, 62, 67
Free Software Foundation, 23
Functions, 135
even/odd, 453
Functions, examples, 136
Gauss, Karl Friedrich, 7
Gaussian algorithm, 26
Gaussian elimination, 7, 9, 490
(See A = LU and PA = LU.)
Gauss-Jordan elimination, 9, 26, 32
General solution, 46
General solution of system, 67, 123
General systems of linear equations, 6
General vector spaces, 160
Generalized eigenvalue problem, 399
Geometric figure, 476
Geometric interpretation of vectors, 94
Geometric multiplicity, 373
Geometric progression, 576
Gerschgorin
column discs, 534
row discs, 533
stronger, 562
Gerschgorin, Semyon Aranovich, 533
Gerschgorin’s Theorem, 533
Glancing ball, 424
GNU General Public License, 23
Golub, Gene Howard, 501
Gram, Jorgen Pedersen, 447
nonsingular, 448
Gram-Schmidt algorithm,
modified, 437
unnormalized, 435
Gram-Schmidt process, 433
Hadamard’s Inequality, 457
Hamilton, William Rowan, 469
Harmonic mean, 577
Height of triangle, 253
Hermitian matrix, 459
eigenvalues, 463
HITS algorithm, 477
Homogeneous linear equations, 60, 62, 230
Homogeneous polynomial, 472
Hooke, Robert, 51
Hooke’s Law, 51
Hub, 545
Hyperbola, 476
Identity map, 146
Identity matrix, 193
If and only if, 565
Ill-conditioned, 87
Imaginary number, 581
Imaginary part, 581
Immigration, 545
Implications, 565
Inconsistent equations, systems, 4, 47
Independent. (See Linear independent.)
Index, 170
Indexed sets of vectors, 72
Inequalities, 206
Infinite dimensional, vector space(s), 109, 319–320
Infinite sequences, [x1,x2,x3,…], 165
Infinitely many solutions, 5
Infinity norm, 539
Initial condition, 546
Injective, 137
Inner product, 107, 405, 429, 432, 453, 457, 483, 485
properties, 405
weighted, 407
vector spaces, 404
Input-output matrix, 178
Integer data, 12
Interchanges, 16
Intermediate demand, 549
Intermediate demand vector, 179
Interpolation, 231
Interpreting linear systems, 42
Intersection, subspaces, 307
Intuitive interpretation, 21
Inverse, computing, 226
Inverse image f−1[S]/T−1[W], 294, 308
Inverse matrix,
2 × 2, 235
2 × 2 block, 508
left/right, 227
products matrices, 224
using determinants, 276
Invertible Matrix Theorem, 230–231
Invertible (nonsingular), 222, 228
Irrational, 578
Isomorphic, 377
, 316
, 318
Isomorphic vector spaces, 317
Isomorphism, 317
Iterative algorithms/methods, 324
computing eigenvalues, 542–544
conjugate gradient method, 530
convergence properties, 541
Gauss-Seidel method, 528
Jacobi iterative method, 526
Richardson’s method, 525
SOR method, 529
Kernel, Ker(A), 60, 144, 231, 299
Kernel of a linear transformation, Ker(T), 295, 329
Kirchhoff, Gustav Robert, 101
Kirchhoff’s First Law, 101
Kirchhoff’s Second Law, 101, 214
Latent roots, 368
Law of Cosines, 416
LDLT -factorization/decomposition, 497
Leading 1 (pivot), 17
Leading nonzero, 20
Least squares problem, 127–129, 133, 439–445, 457
Left inverse, 216, 220, 227, 230
Legendre, Adrien-Marie, 361
Legendre polynomials, 313, 335
Length, 407
Length of vector, 91
Leontief Closed Model, 177
Leontief systems, 177–180, 548–550
Leontief, Wassily, 177
Lexicographical order, 560
Line, 95
described parametrically, 115, 117
passing through origin, 112, 114
Line through point, parallel to vector, 95
Linear algebra, 1
Linear combination of columns, 43
Linear combination of vectors, 34, 92
(See Systems linear equations.)
Linear function, 274
Linear inequalities, 206
Linear least-squares solution, 127–129, 133, 439–445, 457
parabolic/quadratic, 457
Linear maps, using matrices, 142
Linear ordinary differential equations, 50
Linear programming, 207
Linear system, 230
Linear transformation(s), 134, 139, 145, 200, 230, 295, 347, 362, 371
composition, 156
dimension, 332
effects, 145
eigenvalues/eigenvector, 371
expansions, 156
identity, 159
injective, 144
linearly independent set, 176
orthogonal projection, 156, 159
parallel to x-axis, 160
parallel to y-axis, 160
projections, 156
reflection, 148, 149, 155, 159
rotate points, 149
scaling, 160
sequencing, 160
set, 326
surjective, 144
zero, 159
Linearly dependent, indexed sets, 73, 79, 169, 171–173
Linearly dependent, set of vectors, 73, 79
Linearly dependent sets, 61, 167, 169
Linearly independent, indexed sets, 73, 79, 169, 171–173
Linearly independent, set of vectors, 73, 79
Linearly independent orthogonal set, 411
Linearly independent set of vectors, algorithm, 75
Linearly independent sets, 61, 167, 169
Lines, 112
in , 115
in , 122
LLT-factorization/decomposition, 494, 498
parametric form, 120
same, 117
Long-term behavior, dynamical system, 381–385
Lower triangular matrix, 194
LU-factorization/decomposition, 224, 489
LU-factorization/decomposition of PA, 492
Magnitude, 407
Manufactured goods, 551
Maple, 25, 52, 53, 234, 257, 282, 391, 292, 451, 480, 513, 514, 516, 517, 532
www.maplesoft.com, 23
Mapping, 134
set f[U], 293
a vector space into itself, 351
Mathematica, 25, 53, 234, 257, 391, 392, 393, 394, 452, 480, 513, 515, 516, 518
www.wolfram.com, 23
Mathematical induction, 567
Mathematical software, 23, 52, 234, 259, 282, 391, 451, 480, 512, 552
MATLAB, 24, 52, 234, 257, 391, 393, 394, 451, 480, 513, 514, 515, 517, 552, 553, 554
www.mathworks.com, 23
addition, 186
adjacency, 485
cofactor, 266
decomposition, 487
diagonal, 193
equality, 187
factorization, 487
form(s), 43
Hermitian, 458
inverse, 216
lower/upper triangular, 194
m × n matrices, 7
market, 186
multiplication, 188
n × n matrix, 7
noncommutativity, 198–199, 208–209
norm, 407
orthogonal, 466
pencil, 399
positive definite, 494
post-multiplication, 191
pre-multiplication, 191
scalar multiplication, 186
skew-Hermitian, 398
special structures, 194
square, 221
square root, 498
trace, 307
unitary, 466
upper triangular, 194
Matrix-matrix
Associative Law, 199
Matrix-vector product, Ax, 38
Median triangle, 575
Method of contradiction, 567
Method of least squares, 127
Midpoint, 131
Migration, 545
Minimizing ||Ax − b||, 441
Minimum daily requirements, diet, 205
Minors, 264
expansion using row i/column j, 365, 366
Models in economic theory, 176
Modified Gram-Schmidt process, 437
Modulo arithmetic, 355
Modulus, 582
de Moivres, Abraham, 582
de Moivres’ Theorem, 582
Moment of weights, 96
Multiplication by scalars, 89
unit scalar times vector, 161
Multiplication of matrices, 188
Multiplicity, 373
Multiplier, 11
Mutually orthogonal vectors, 411
, 165
Natural basis, 345
Natural numbers, , 165
Natural ordering, 560
n-dimensional Euclidean
vector space, 90
inner product space, 405
Network problem(s)
Neumann series, 550
Nodes, 99
No pivot positions, 66
Noncommutativity, matrix multiplication, 198, 208
Noninvertible (singular) matrix, 222, 227
Nonnegative definite matrix, 495
Nonnegative vectors, 548
Nonsingular (invertible) matrix, 222, 228
Nonsquare matrices, 227
Nontrivial solution, 60
Nonzero determinant, 251
Nonzero eigenvalue, 229
Nonzero rows, 66
Nonzero singular value, 231
Normal equation, 441, 443, 451
Normal to plane, 121
Normal vector, 451
Normalization, 410
Norm of a vector/matrix, 368
infinity, 539
Euclidean, 407
n-tuple, 88
Null space, Null(A), 60, 231, 299, 418
Null(T), 329
Number of basis vectors, 319
Number of elements in set T, #(T), 337
Octave, 23
Odd function, 454
Off-diagonal elements, 193
Ohm, George Simon, 101
Ohm’s Law, 101
One degree of freedom, 21
Open sector, 549
Optimal solution, 212
Ordered basis, 315
Ordered set, 342
complement, 416
complement of row space, 418
matrix, 186, 400, 420, 466, 582
mutually, 411
properties, 414
subspace, 422
SVD, 501
(See Gram-Schmidt process.)
Orthonormal set of vectors, 412, 419, 432
Orthonormality, 91, 121, 419, 431
Overdetermined system, 133, 441, 444, 510
(See Linear least squares solution.)
Overrelaxation (SOR), 529
, 291
, 349
PA = LU, 492
PageRank, 478
PageRank algorithm, 478
Parallel lines, 116
distinct, 118
same, 117
Parallel matrix-matrix multiplication algorithm, 522–523
Parallelepiped, 260
Parallelogram, calculating area, 256, 260
Parallelogram, diagonal vector, 34
Parallelogram Law, 428
Parameters/free variables, 56, 62, 67
Parametric line, 114–115, 117, 120
Parametric plane, 120
Partial-fraction decomposition, 126, 131, 575
Partial pivoting, 24
sums, 574
Partitioned matrices, 504
Perpendicular. (See Orthogonal.)
Perpendicular bisector, 131
columns, 324
row, 11
Pivoting strategy, 32
determinants, 276
parametric form, 120
in , 273
through the point p, 450
Point closest to a hyperplane, 450–451
Point closest to righthand side b, 447
Poisson equation, 560
Polar coordinates, 582
Pole, 368
Political affiliation problem, 402
Polynomials
Chebyshev, 336, 345, 349, 359, 361
interpolation, 231
natural basic, 174
Population migration, 545
Positive definite matrix, 401, 520
test, positive definiteness, 494
Positive integers, , 136
Postmultiplication by columns, 191, 202
Power method, 544
Powers of matrices, 376, 471–472
Predator-prey simulation, 125
Premultiplication by rows, 191, 202
Price vector, 177
Prime, twin, 563
Principal minors, 494
Product of matrices, 192
(See Matrix-matrix multiplication.)
Product upper triangular matrices, 195
Production levels, 548
Production vector, 549
orthogonal, 147, 413, 430, 446
vector onto subspace, 422
Proofs, 563
questionable, 572
Proper factor, 136
Properties of
Vector Spaces, 90
Pythagorean
Law, 430
Theorem, 409
triangle, 575
QR algorithm. (See Gram-Schmidt process.)
QR-decomposition/factorization, 499
Quadratic forms, 472
Quadratic least squares, 457
Quantified Assertion, 572
Quantifiers, 571
Questionable proofs, 572–573, 577–579
Quotient, 582
, 165
, 165
Range of
function f, Range(f), 135
linear transformation, Range(T), 298, 329
Rank and row equivalence, 230
Rank-Nullity Theorem, 331
Rational, 578
Real n-space, 90
Real part, 581
Real-valued function of real variable, 135
Recurrence relation, 579
Red-black ordering, 561
Reduced row echelon form, 17, 26
algorithm, 26
ReducedRowEchelonForm (A),19
rref (A),19
rrefmovie,19
Uniqueness, 63
Reduced system, 507
Reduction, 147
Repeller, 384
Replacement, elementary matrix, 201
Replacement operator, 8, 9, 16, 25, 32, 244, 301
Residual, 127
Restriction, 308
Reversible, 10
Review determinant: notation, properties, 282
Richardson iterative method, 525, 541
Right inverse, 219, 220, 227, 230
Rotate points, 149
counterclockwise by φ, 150
Row echelon form, 20, 230, 355
Row equivalent, 10, 11, 44, 49, 301
systems, 44
Row index, 17
Row operations, 11
, 201
RowOperation,25
Row rank, 332
Row reduce, RowReduce [A],19
Row reduction algorithm/process, 9, 26
Row replacement operation, 8–9, 16, 25, 25, 27, 201, 244
elementary matrices, 201
Row swapping, 245
Rows, 17
Rural area, 545
Saddle point, 384
Scalar multiplication, 160, 187
_scale,26
Scale, elementary matrix, 201
Scale operation, 9
Scaled, 526
Scaling phase, 32
Schilder’s factorization, 522
Schmidt, Erhard, 433
Schur complement, 508
Schur Decomposition Theorem, 502
Schur, Issai, 502
Schwarz, Hermann Amandus, 408
Searching, World Wide Web, 477
Seidel, Philip Ludwig von, 528
Self-adjoint
mapping, 462
Set difference, 570
Set of all mappings, VS, 185, 488
Sherman-Morrison formula, 510
Sign of flow across battery, 102
Signals in electrical engineering, 165
Similar matrices A B, 352, 372
Similarity, 363
Singular (noninvertible) matrix, 222
Singular-value decomposition (SVD), 501
Sink, 384
Skew-Hermitian, 482
Skew-symmetric matrix, 197, 286, 482
Slope, 2
Slope-intercept, 107
Smoothing data, 152
Software, mathematical, www.netlib.org, 32
Solutions
extraneous, 159
infinitely many, 5
none, 5
nontrivial, 60
Solving 2 × 2 block system, 506
Solving linear systems using LU-factorization, 489
Solving systems of linear equations, 1
with a left inverse, 216
with a right inverse, 219
SOR method, 529
Source, 384
Space. (See Vector space.)
Span of set of columns, 48
Span, Span{a1, a2,…,an},39
Spanning set, 338
Sparse matrix, 524
Special matrices, 193
Spectral radius, 558
Spectral Theorem, 465
Matrix Version, 466
Spectrum, eigenvalues, 558
Spiral, 384
Square root of matrix, , 498
Stable system, 96
Standard inner product, 406, 459
Standard unit vectors, 311
in , 459
in , 92
Statistical trend analysis, 108
Stencils, 531
Strict diagonal dominant, 558
Strong induction, 568
Submatrices, 504
Subsets, 570
Subspace(s), 289
Inner-product spaces, 422
Intersection, 307
, 291
point closest to projected point, 450
projection of vector onto subspace, 445, 448
, 166
Row(A), 300
Sum, 307
Union, 307
Subspaces in inner product spaces, 422
Suburban area, 545
Successive overrelaxation (SOR) method, 529
Summation by parts, 578
Surface tension, 128
SVD, 501
Swap, elementary matrix, 201
Swapping, 245
Sylvester criterion, 494
Sylvester, James Joseph, 189, 190
Symmetric matrix, 197
Symmetric real matrix, eigenvalues, 463
Systems of linear differential equations, 388
Systems of linear equations, 1, 4
equivalent forms of Ax = b,43
iterative methods for solving, 524
_swap, 26
Swap, elementary matrix, 201
Swapping, 295
Test for positive definiteness, 494
Theorems on vector spaces, 162–163
Traffic circle, 111
Traffic flow problems, 99, 107–111
Trajectory, dynamical system, 381
spiral, 383
Transformation, 134
(See Linear Transformation.)
Transformation on geometrical figures, 150
Transformation matrix, 145
Translation, 114
Transition matrix, 343
Translation, 114
Transpose matrix, 196
determinant, 273
Triangle, calculating area, 256
Triangle in a rectangle, 255
Triangular form, 8
Triangular matrices, 244
Truth table, 569
Tuple, n, 88
Twin primes, 563
Two degrees of freedom, 21
Two-point form, 3
Union, 570
Union, subspaces, 307
Uniqueness Reduced Row Echelon Form, 63
Unit consumption vector, 550
Unit vectors, 410
in , 36
in , 40
Unitarily similar, 461
A = UBUH, 468
Universal quantifier, 571
Unknown vector, 42
Unnormalized Gram-Schmidt algorithm, 435
Updating formula, 528
Upper triangular matrix, 194
Upper bi-diagonal, 153
Upper tri-diagonal, 153
Vandermonde, Alexandre-Théophile, 279
Vandermonde matrix, 279
basic/free, 67
(See Change of variables.)
Vector addition, 33, 89, 94, 160
Vector dot product, 91, 107, 192, 406
Vector intersection, 292
Vector-matrix product, 203
Vector of unknowns, 6
over field F, 164
real, 164
Vector subspace(s), 289
column space, 40, 48, 230, 324, 332,
Vector subspace, polynomials , 291
Vector sum, 292
Vector-matrix product, 203
Vectors and matrices, 32
Venn diagrams, 571
Vertical intercepts, 107
Voltage Law, 101
Volumes, calculating, 253
18.191.189.23