A

A = LDL^T, 497

A = LL^T, 494

A = LU, 224, 489

A = PBP⁻¹, similarity, 352

A = PDP⁻¹, diagonalizable, 372

A^k = PD^kP⁻¹, powers of a matrix, 376

A = PDQ, SVD-decomposition, 501

A = PQP⁻¹, transition, 354

A = QR, 499

A = UBU^H, unitarily similar, 468

A = UDU^H, Spectral Theorem: Matrix Version, 466

A = UTU^H, Schur decomposition, 502

Abel-Ruffini Impossibility Theorem, 584

Addition of

matrices, 187

vectors, 33

Adjacency matrix, 477, 485

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

Altitude, triangle, 253, 575

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

work estimate, 267–268

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

Augmented matrix, 7, 12

Average, 106

Averaging, 152, 153

Axioms, 90, 99, 160–163

_axpy, 26, 32

B

-coordination, 342

Back substitution, 32, 490

Backward dynamical system, 401

Backward elimination phase, 26, 32

Bacteria nutrition problem, 30

Balanced weights, 96

Ball, glancing, 424

Base of triangle, 255

Basic/free variables, 63, 67

Basic Linear Algebra

Subprograms (BLAS), 26, 32, 113

Basic properties of vector spaces , 90

Basis

for vector space, 311, 324

orthogonal basis, 412, 435

orthonormal basis, 412, 421, 435

standard vectors, 311

Battery, sign for flow, 102

Bending beam, 50

Best approximate solution, 440, 444

Bijective, 138, 316

Binomial coefficient, , 577

Binomial Theorem, 577

Biological species, 133

Bivariant interpolation, 240

BLAS, 26, 32

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

C

, 184

C[a, b], 407, 419

, 406, 428, 459, 463

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

Center of mass, 97, 106

Centroid, 106

Chain rule, 151–152

Change of basis, 343–347

Change of variables, 474–477

Changing coordinates, 343

Characteristic equation/values, 368, 369, 377

Chebyshev, Pafnuty Lvovich, 345

Chebyshev polynomials, 336, 345, 349, 359, 361

Chemistry, 79, 84

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

Codomain, 135, 329

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

Column space, Col(A), 39, 300

Col(A^T), 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

Component(s), 89, 426

Componentwise, 89

Composition of

linear functions/mapping, 151

linear transformations, 200

Computing inverse, 226

using determinants, 276

Conic section, 87

Conjugate, 404, 582

Conjugate gradient algorithm/method, 530–531

Conjugate transpose, 461

Consistent systems, 4, 47

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

Coordinate vector, , 315, 342

Coordination, 342

Cosines, Law, 415

Cramer, Gabriel, 275

Cramer’s Rule, 275–277

Current Law, 101

Curve fitting, 127–129, 444–445

(See Linear least squares.)

Cycle, 384

D

, 413, 421, 498

Dangerous pitfalls, 208, 209

(See Cautions.)

Data motion, 17

Data smoothing, 152

Decomposition, (See Factorization.)

Deductive reasoning, 563, 564

Demand vector, 549

Demographic problems, 545

Denial of a quantified assertion, 572

Dependent. (See Linearly dependent.)

Determinant(s), 231, 243

2 × 2 matrices, 248, 268

3 × 3 matrices, 269

algorithm without scaling, 249

calculating inverses, 276–278

cofactors, 264

column expansion, 266–267

computing, 246

direct calculation methods, 268–269

finding eigenvalues, 369

linear function, 274

minors, 264, 272

nonzero, 251

product of matrices, 271

properties, 244, 264, 270

Properties I, II, III, 244

replacement, 244

review, 282

row expansion, 265–266

scaling, 244

swapping, 245

transpose, 273

triangular matrix, 244

without scaling, 249

zero, 250

Diag(d₁,d₂, …,d_n), 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

Domain, 135, 329

Dominant eigenvalue, 554

Dot product, 91, 107, 192, 406

(See Inner product.)

Dynamical systems, 380, 401

analysis, 384

backward/forward, 401

in , 384

E

Echelon form, 17–20

Economic models, 385

Eigenproblem, 368

Eigenspace, 398

Eigensystem, without determinants, 389

Eigensystem(s), 367

Eigenvalues, 231, 368, 371

characteristic equation/polynomial, 369, 377

complex conjugate pairs, 380

distinct, 374

generalized problem, 399

Hermitian matrix, 459, 463

linear operator, 371

matrix, 368

positive definite matrix, 401, 494

powers of matrix, A^k, 376–377, 469

real symmetric matrix, 463

self-adjoint operator, 462

Eigenvectors, 361, 371

linear operator, 371

matrix, 368

Electrical circuits/networks, 101

Elementary matrices, 201, 224

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 matrices, 11–12

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

F

Factorization, 487

LDL^T , 497

LL^T, 494, 497

LU, 225, 489

LU of PA, 492

QR, 499

SVD, 501

xⁿ − yⁿ, 574

Feeding bacteria, 22

Fibonacci sequence, 363, 401

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 among sectors, 179–180

Flow Axiom, 99

Flow problems, 108–112

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

Fundamental Theorem of Algebra, 7, 583

G

Gauss, Karl Friedrich, 7

Gaussian algorithm, 26

Gaussian elimination, 7, 9, 490

(See A = LU and PA = LU.)

Gauss-Jordan elimination, 9, 26, 32

Gauss-Seidel method, 529, 542

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

Gram matrix, 445–448

nonsingular, 448

Gram-Schmidt algorithm,

modified, 437

unnormalized, 435

Gram-Schmidt process, 433

Graphs and networks, 99–111

Grid problem, 110, 111

H

Hadamard’s Inequality, 457

Hamilton, William Rowan, 469

Harmonic mean, 577

Height of triangle, 253

Hermite, Charles, 406, 459

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

Hyperplane, 123, 458

I

Identity map, 146

Identity matrix, 193

If and only if, 565

Ill-conditioned, 87

Image(s), 135, 293

Imaginary number, 581

Imaginary part, 581

Immigration, 545

Implications, 565

Implies, 4, 47

Inconsistent equations, systems, 4, 47

Independent. (See Linear independent.)

Index, 170

Indexed set, 169–172

Indexed sets of vectors, 72

Induction, 567–568

Industry, 177, 386

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

standard, 406, 459

weighted, 407

vector spaces, 404

Input-output matrix, 178

Integer data, 12

Intercept(s), 2, 107

Interchanges, 16

Intermediate demand, 549

Intermediate demand vector, 179

Interpolation, 231

Interpreting linear systems, 42

Intersection, 292, 570

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

J

Jacobi iterative method, 526

Jordan, Camille, 9

Jordan, Wilhelm, 9

K

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

Kronecker delta, 18, 193, 214, 419

Kronecker, Leopold, 18, 193

L

Latent roots, 368

Law of Cosines, 416

LDL^T -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 Open Model, 179, 549

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,2,46

Linear algebra, 1

Linear combination of columns, 43

Linear combination of vectors, 34, 92

Linear equation(s), 2, 3

(See Systems linear equations.)

Linear function, 274

Linear inequalities, 206

Linear least-squares solution, 127–129, 133, 439–445, 457

parabolic/quadratic, 457

Linear mapping, 139, 174

Linear mapping, x Ax, 46–47

Linear maps, using matrices, 142

Linear operators, 371, 462

Linear ordinary differential equations, 50

Linear programming, 207

Linear system, 230

Linear transformation(s), 134, 139, 145, 200, 230, 295, 347, 362, 371

codomain, 135, 329

composition, 156

contraction, 156, 159

dilation, 147, 156, 159

dimension, 332

domain, 135, 329

effects, 145

eigenvalues/eigenvector, 371

expansions, 156

identity, 159

injective, 144

kernel, 295, 299, 329

linearly independent set, 176

null space, 299, 329

orthogonal projection, 156, 159

parallel to x-axis, 160

parallel to y-axis, 160

projections, 156

range, 135, 298, 329

reflection, 148, 149, 155, 159

rotate points, 149

rotation, 146, 150, 156, 159

scaling, 160

sequencing, 160

set, 326

shear, 148, 156, 160

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

identical, 5, 118

in , 95, 112

in , 115

in , 122

intersecting, 5, 118

LL^T-factorization/decomposition, 494, 498

parallel, 5, 118

parallel and distinct, 5, 118

parametric form, 120

same, 117

Localize eigenvalues, 533–539

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

Matrices, 6, 186

Matrix, 11, 32, 186

addition, 186

adjacency, 485

augmented, 7, 43

block, 487, 504

coefficient, 7, 42, 230

cofactor, 266

decomposition, 487

diagonal, 193

diagonalizable, 372, 540

echelon form, 17–20

elementary, 201, 224

equality, 187

factorization, 487

form(s), 43

Hermitian, 458

identity, 18, 193

inverse, 216

invertible, 222, 228

lower/upper triangular, 194

m × n matrices, 7

market, 186

multiplication, 188

n × n matrix, 7

noncommutativity, 198–199, 208–209

noninvertible, 222, 228

nonsingular, 222, 228

norm, 407

normal, 440–444, 511

operations, 186–192

orthogonal, 466

pencil, 399

permutation, 487, 492

positive definite, 494

post-multiplication, 191

pre-multiplication, 191

projection, 147, 156

reflection, 148, 155

rotation, 146, 149–151

scalar multiplication, 186

similar, 352–354

singular, 222, 228, 231

skew-Hermitian, 398

skew-symmetric, 197, 398

special structures, 194

square, 221

square root, 498

symmetric, 197, 398

trace, 307

transitive, 353, 355

transpose, 33, 196

unitary, 466

upper triangular, 194

Vandermonde, 279–280, 287–288

Matrix-matrix

Associative Law, 199

multiplication, 188, 199

product, 68, 202

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

Monomials, 173, 291

de Morgan Laws, 570, 575

Multiplication by scalars, 89

unit scalar times vector, 161

Multiplication of matrices, 188

Multiplicity, 373

Multiplier, 11

Mutually orthogonal vectors, 411

N

, 165

Natural basis, 345

Natural numbers, , 165

Natural ordering, 560

n-dimensional Euclidean

vector space, 90

inner product space, 405

Network problem(s)

electrical, 101, 108–111

traffic, 99, 107–111

Neumann series, 550

Nodes, 99

No pivot positions, 66

No solution, 5, 57

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 matrix, 483, 486

Normal to plane, 121

Normal vector, 451

Normalized, vector, 391, 408

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

Nullity, 329, 331

Number of basis vectors, 319

Number of elements in set T, #(T), 337

0

Octave, 23

Odd function, 454

Off-diagonal elements, 193

Ohm, George Simon, 101

Ohm’s Law, 101

One degree of freedom, 21

One-to-one, 137, 230

Onto, 138, 230

Open sector, 549

Optimal solution, 212

Ordered basis, 315

Ordered set, 342

Origin, 97, 112

Orthogonal, 411, 412, 466

basis, 412, 435

complement, 416

complement of row space, 418

matrix, 186, 400, 420, 466, 582

mutually, 411

projection, 147, 413, 446

properties, 414

subspace, 422

SVD, 501

(See Gram-Schmidt process.)

Orthonormal set of vectors, 412, 419, 432

basis, 419, 482

Orthonormality, 91, 121, 419, 431

Outer product, 213, 461

Overdetermined system, 133, 441, 444, 510

(See Linear least squares solution.)

Overrelaxation (SOR), 529

P

, 291

, 174, 175

, 349

, 165, 317

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

Permutation matrix, 487, 492

Perpendicular. (See Orthogonal.)

Perpendicular bisector, 131

Pivot(s), 17, 230

columns, 324

positions, 17, 48, 64–66, 230

row, 11

Pivoting strategy, 32

Plane(s), 95, 118–123

determinants, 276

parametric form, 120

in , 273

in , 122–123

through the origin, 120–121

through the point p, 450

Point closest to a hyperplane, 450–451

Point closest to righthand side b, 447

Points in , 88–89

Point-slope form, 2, 107

Poisson equation, 560

Polar coordinates, 582

Pole, 368

Political affiliation problem, 402

Polynomials

characteristics, 369, 377

Chebyshev, 336, 345, 349, 359, 361

interpolation, 231

Legendre, 313, 361

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

Projection, 156, 429

orthogonal, 147, 413, 430, 446

vector onto subspace, 422

vector onto vector, 413–414

Projection matrix, 414–415

Proofs, 563

questionable, 572

Proper factor, 136

Properties of

determinants, 244, 270

least squares, 440–445

Vector Spaces, 90

Pythagoras of Samos, 408, 564

Pythagorean

Law, 430

Theorem, 409

triangle, 575

triple, 564, 574

Q

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

R

, 115–117

, 75, 89, 230–231, 317, 412

, 165

, 165

, 318, 350

, 187, 319

Random matrix, 31, 241

Range of

function f, Range(f), 135

linear transformation, Range(T), 298, 329

Rank, Rank(A), 63, 230, 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

Reciprocal, 565, 582

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

Reflections, 148, 149, 456

Reflexive, 11, 317, 353, 355

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

Righthand-side vector, 7, 42

Rotate points, 149

Rotation, 146, 156

counterclockwise by φ, 150

Roundoff error(s), 24, 59

Row echelon form, 20, 230, 355

Row equivalence, 11, 49

Row equivalent, 10, 11, 44, 49, 301

matrices, 10, 11

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 space, Row(A), 300, 304

Row swapping, 245

Row vectors, 231, 405

Rows, 17

Rural area, 545

S

Saddle point, 384

Scalar, 34, 89, 91, 164

Scalar multiplication, 160, 187

Scale, 16, 25, 44, 301

_scale,26

Scale, elementary matrix, 201

Scale operation, 9

Scaled, 526

Scaling, 15, 244

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

Sectors, 177, 185, 549

Seidel, Philip Ludwig von, 528

Self-adjoint

mapping, 462

operator, 462, 481

Set difference, 570

Set of all mappings, V^S, 185, 488

Shear, 148, 156

Sherman-Morrison formula, 510

Sign of flow across battery, 102

Signals in electrical engineering, 165

Similar, 352, 468

Similar matrices A B, 352, 372

Similarity, 363

Singular (noninvertible) matrix, 222

Singular-value decomposition (SVD), 501

Singular values, 231, 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

trivial, 47, 60

unique, 5, 229, 230

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, 39, 92

Span of set of columns, 48

of vectors, 39, 92

Span, Span{a₁, a₂,…,a_n},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 matrix, 193, 221

Square root of matrix, , 498

Stable system, 96

Staircase pattern, 18, 20

Standard inner product, 406, 459

Standard unit vectors, 311

in , 459

in , 92

in , 142, 311

Statistical trend analysis, 108

Stencils, 531

Strict diagonal dominant, 558

Strong induction, 568

Submatrices, 504

Subsets, 570

Subspace(s), 289

basis, 312, 323–324, 326

Col(A), 39, 48, 230, 300

Inner-product spaces, 422

Intersection, 307

Null(A), 60, 231

, 291

point closest to projected point, 450

projection of vector onto subspace, 445, 448

, 166

, 133, 311

, 187, 319

Row(A), 300

Sum, 307

Union, 307

Subspaces in inner product spaces, 422

Suburban area, 545

Successive overrelaxation (SOR) method, 529

Sum of subspaces, 292, 307

Summation, 4, 43

Summation by parts, 578

Surface tension, 128

Surjective, 138, 144, 230

SVD, 501

Swap, 16, 25, 32, 44

Swap, elementary matrix, 201

Swapping, 245

Sylvester criterion, 494

Sylvester, James Joseph, 189, 190

Symmetric, 11, 317, 353, 355

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

homogeneous, 60, 62, 320

_swap, 26

Swap, elementary matrix, 201

Swapping, 295

T

Taussky-Todd, Olga, 575–576

Test for positive definiteness, 494

Theorems on vector spaces, 162–163

There exists, 216, 307, 571

Trace, 216, 307

Traffic circle, 111

Traffic flow problems, 99, 107–111

Traffic Problem I, 100, 107

Trajectory, dynamical system, 381

spiral, 383

Transformation, 134

(See Linear Transformation.)

Transformation on geometrical figures, 150

Transformation matrix, 145

Translation, 114

Transition matrix, 343

Transitive, 11, 317, 353, 355

Translation, 114

Transpose matrix, 196

determinant, 273

Triangle, calculating area, 256

Triangle Inequality, 409, 576

Triangle in a rectangle, 255

Triangular form, 8

Triangular matrices, 244

Triangular numbers, 576, 580

Trivial solution, 47, 60, 230

Truth table, 569

Tuple, n, 88

Twin primes, 563

Two degrees of freedom, 21

Two-point form, 3

U

Union, 570

Union, subspaces, 307

Unique solution, 5, 230

Uniqueness Reduced Row Echelon Form, 63

Unit consumption vector, 550

Unit vectors, 410

in , 36

in , 40

in , 142, 311

Unitarily similar, 461

A = UBU^H, 468

Unitary matrix, 420, 466, 468

Universal quantifier, 571

Unknown(s), 3, 6, 7

Unknown vector, 42

Unnormalized Gram-Schmidt algorithm, 435

Updating formula, 528

Upper triangular matrix, 194

Upper bi-diagonal, 153

Upper tri-diagonal, 153

V

Vandermonde, Alexandre-Théophile, 279

Vandermonde matrix, 279

determinant, 280, 287

Variables, 3, 4

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

Vector(s), 33, 89, 91

Vector space(s), 88, 160, 289

over field F, 164

, 90, 161

real, 164

Vector subspace(s), 289

column space, 40, 48, 230, 324, 332,

isomorphic, 317, 320

kernel, 60, 144, 231, 299

null space, 231, 418, 299

row space, 231, 232, 418

Vector subspace, polynomials , 291

Vector sum, 292

Vector-matrix product, 203

Vectors, 1, 32, 88, 160, 164

Vectors and matrices, 32

Venn diagrams, 571

Vertical intercepts, 107

Voltage Law, 101

Volumes, calculating, 253

W

Weak diagonal dominant, 558

Weak induction, 568

Web page links, 478

Weights, 96

Woodbury formula, 509

Woodbury matrix identity, 509

Work and forces, 423

Work estimate, 267

World Wide Web Searching, 477

X

x-vector, 6, 33

Y

Young, David M., Jr., 529–530

Young’s equation, 529

Z

, 165, 187

Zero determinant, 250

Zero eigenvalues, 231

Zero matrix, 197

Zero null space, 280

Zero solution, 47, 60

Zero vector, 91, 229, 231, 280