Index

A (appendix), 1
A (archetype), 2
A (definition), 3
A (notation), 4
A (part), 5
AA (Property), 6
AA (subsection, section WILA), 7
AA (theorem), 8
AAC (Property), 9
AACN (Property), 10
AAF (Property), 11
AALC (example), 12
AAM (Property), 13
ABLC (example), 14
ABS (example), 15
AC (Property), 16
ACC (Property), 17
ACCN (Property), 18
ACF (Property), 19
ACM (Property), 20
ACN (example), 21
additive associativity
    column vectors
        Property AAC, 22
    complex numbers
        Property AACN, 23
    matrices
        Property AAM, 24
    vectors
        Property AA, 25
additive closure
    column vectors
        Property ACC, 26
    complex numbers
        Property ACCN, 27
    field
        Property ACF, 28
    matrices
        Property ACM, 29
    vectors
        Property AC, 30
additive commutativity
    complex numbers
        Property CACN, 31
additive inverse
    complex numbers
        Property AICN, 32
    from scalar multiplication
        theorem AISM, 33
additive inverses
    column vectors
        Property AIC, 34
    matrices
        Property AIM, 35
    unique
        theorem AIU, 36
    vectors
        Property AI, 37
adjoint
    definition A, 38
    inner product
        theorem AIP, 39
    notation, 40
    of a matrix sum
        theorem AMA, 41
    of an adjoint
        theorem AA, 42
    of matrix scalar multiplication
        theorem AMSM, 43
AHSAC (example), 44
AI (Property), 45
AIC (Property), 46
AICN (Property), 47
AIF (Property), 48
AIM (Property), 49
AIP (theorem), 50
AISM (theorem), 51
AIU (theorem), 52
AIVLT (example), 53
ALT (example), 54
ALTMM (example), 55
AM (definition), 56
AM (example), 57
AM (notation), 58
AM (subsection, section MO), 59
AMA (theorem), 60
AMAA (example), 61
AME (definition), 62
AME (notation), 63
AMSM (theorem), 64
ANILT (example), 65
ANM (example), 66
AOS (example), 67
Archetype A
    column space, 68
    linearly dependent columns, 69
    singular matrix, 70
    solving homogeneous system, 71
    system as linear combination, 72
archetype A
    augmented matrix
        example AMAA, 73
Archetype B
    column space, 74
    inverse
        example CMIAB, 75
    linearly independent columns, 76
    nonsingular matrix, 77
    not invertible
        example MWIAA, 78
    solutions via inverse
        example SABMI, 79
    solving homogeneous system, 80
    system as linear combination, 81
    vector equality, 82
archetype B
    solutions
        example SAB, 83
Archetype C
    homogeneous system, 84
Archetype D
    column space, original columns, 85
    solving homogeneous system, 86
    vector form of solutions, 87
Archetype I
    column space from row operations, 88
    null space, 89
    row space, 90
    vector form of solutions, 91
Archetype I:casting out vectors, 92
Archetype L
    null space span, linearly independent, 93
    vector form of solutions, 94
ASC (example), 95
augmented matrix
    notation, 96
AVR (example), 97

B (archetype), 98
B (definition), 99
B (section), 100
B (subsection, section B), 101
basis
    columns nonsingular matrix
        example CABAK, 102
    common size
        theorem BIS, 103
    crazy vector apace
        example BC, 104
    definition B, 105
    matrices
        example BM, 106
        example BSM22, 107
    polynomials
        example BP, 108
        example BPR, 109
        example BSP4, 110
        example SVP4, 111
    subspace of matrices
        example BDM22, 112
BC (example), 113
BCS (theorem), 114
BDE (example), 115
BDM22 (example), 116
best cities
    money magazine
        example MBC, 117
BIS (theorem), 118
BM (example), 119
BNM (subsection, section B), 120
BNS (theorem), 121
BP (example), 122
BPR (example), 123
BRLT (example), 124
BRS (theorem), 125
BS (theorem), 126
BSCV (subsection, section B), 127
BSM22 (example), 128
BSP4 (example), 129

C (archetype), 130
C (definition), 131
C (notation), 132
C (part), 133
C (Property), 134
C (technique, section PT), 135
CABAK (example), 136
CACN (Property), 137
CAEHW (example), 138
CAF (Property), 139
canonical form
    nilpotent linear transformation
        example CFNLT, 140
        theorem CFNLT, 141
CAV (subsection, section O), 142
Cayley-Hamilton
    theorem CHT, 143
CB (section), 144
CB (theorem), 145
CBCV (example), 146
CBM (definition), 147
CBM (subsection, section CB), 148
CBP (example), 149
CC (Property), 150
CCCV (definition), 151
CCCV (notation), 152
CCM (definition), 153
CCM (example), 154
CCM (notation), 155
CCM (theorem), 156
CCN (definition), 157
CCN (notation), 158
CCN (subsection, section CNO), 159
CCRA (theorem), 160
CCRM (theorem), 161
CCT (theorem), 162
CD (subsection, section DM), 163
CD (technique, section PT), 164
CEE (subsection, section EE), 165
CELT (example), 166
CELT (subsection, section CB), 167
CEMS6 (example), 168
CF (section), 169
CFDVS (theorem), 170
CFNLT (example), 171
CFNLT (subsection, section NLT), 172
CFNLT (theorem), 173
CFV (example), 174
change of basis
    between polynomials
        example CBP, 175
change-of-basis
    between column vectors
        example CBCV, 176
    matrix representation
        theorem MRCB, 177
    similarity
        theorem SCB, 178
    theorem CB, 179
change-of-basis matrix
    definition CBM, 180
    inverse
        theorem ICBM, 181
characteristic polynomial
    definition CP, 182
    degree
        theorem DCP, 183
    size 3 matrix
        example CPMS3, 184
CHT (subsection, section JCF), 185
CHT (theorem), 186
CILT (subsection, section ILT), 187
CILTI (theorem), 188
CIM (subsection, section MISLE), 189
CINM (theorem), 190
CIVLT (example), 191
CIVLT (theorem), 192
CLI (theorem), 193
CLTLT (theorem), 194
CM (definition), 195
CM (Property), 196
CM32 (example), 197
CMCN (Property), 198
CMF (Property), 199
CMI (example), 200
CMIAB (example), 201
CMVEI (theorem), 202
CN (appendix), 203
CNA (definition), 204
CNA (notation), 205
CNA (subsection, section CNO), 206
CNE (definition), 207
CNE (notation), 208
CNM (definition), 209
CNM (notation), 210
CNMB (theorem), 211
CNO (section), 212
CNS1 (example), 213
CNS2 (example), 214
CNSV (example), 215
COB (theorem), 216
coefficient matrix
    definition CM, 217
    nonsingular
        theorem SNCM, 218
column space
    as null space
        theorem FS, 219
    Archetype A
        example CSAA, 220
    Archetype B
        example CSAB, 221
    as null space
        example CSANS, 222
    as null space, Archetype G
        example FSAG, 223
    as row space
        theorem CSRST, 224
    basis
        theorem BCS, 225
    consistent system
        theorem CSCS, 226
    consistent systems
        example CSMCS, 227
    isomorphic to range, 228
    matrix, 229
    nonsingular matrix
        theorem CSNM, 230
    notation, 231
    original columns, Archetype D
        example CSOCD, 232
    row operations, Archetype I
        example CSROI, 233
    subspace
        theorem CSMS, 234
    testing membership
        example MCSM, 235
    two computations
        example CSTW, 236
column vector addition
    notation, 237
column vector scalar multiplication
    notation, 238
commutativity
    column vectors
        Property CC, 239
    matrices
        Property CM, 240
    vectors
        Property C, 241
complex m-space
    example VSCV, 242
complex arithmetic
    example ACN, 243
complex number
    conjugate
        example CSCN, 244
    modulus
        example MSCN, 245
complex number
    conjugate
        definition CCN, 246
    modulus
        definition MCN, 247
complex numbers
    addition
        definition CNA, 248
        notation, 249
    arithmetic properties
        theorem PCNA, 250
    equality
        definition CNE, 251
        notation, 252
    multiplication
        definition CNM, 253
        notation, 254
complex vector space
    dimension
        theorem DCM, 255
composition
    injective linear transformations
        theorem CILTI, 256
    surjective linear transformations
        theorem CSLTS, 257
conjugate
    addition
        theorem CCRA, 258
    column vector
        definition CCCV, 259
    matrix
        definition CCM, 260
        notation, 261
    multiplication
        theorem CCRM, 262
    notation, 263
    of conjugate of a matrix
        theorem CCM, 264
    scalar multiplication
        theorem CRSM, 265
    twice
        theorem CCT, 266
    vector addition
        theorem CRVA, 267
conjugate of a vector
    notation, 268
conjugation
    matrix addition
        theorem CRMA, 269
    matrix scalar multiplication
        theorem CRMSM, 270
    matrix transpose
        theorem MCT, 271
consistent linear system, 272
consistent linear systems
    theorem CSRN, 273
consistent system
    definition CS, 274
constructive proofs
    technique C, 275
contradiction
    technique CD, 276
contrapositive
    technique CP, 277
converse
    technique CV, 278
coordinates
    orthonormal basis
        theorem COB, 279
coordinatization
    linear combination of matrices
        example CM32, 280
    linear independence
        theorem CLI, 281
    orthonormal basis
        example CROB3, 282
        example CROB4, 283
    spanning sets
        theorem CSS, 284
coordinatization principle, 285
coordinatizing
    polynomials
        example CP2, 286
COV (example), 287
COV (subsection, section LDS), 288
CP (definition), 289
CP (subsection, section VR), 290
CP (technique, section PT), 291
CP2 (example), 292
CPMS3 (example), 293
CPSM (theorem), 294
crazy vector space
    example CVSR, 295
    properties
        example PCVS, 296
CRMA (theorem), 297
CRMSM (theorem), 298
CRN (theorem), 299
CROB3 (example), 300
CROB4 (example), 301
CRS (section), 302
CRS (subsection, section FS), 303
CRSM (theorem), 304
CRVA (theorem), 305
CS (definition), 306
CS (example), 307
CS (subsection, section TSS), 308
CSAA (example), 309
CSAB (example), 310
CSANS (example), 311
CSCN (example), 312
CSCS (theorem), 313
CSIP (example), 314
CSLT (subsection, section SLT), 315
CSLTS (theorem), 316
CSM (definition), 317
CSM (notation), 318
CSMCS (example), 319
CSMS (theorem), 320
CSNM (subsection, section CRS), 321
CSNM (theorem), 322
CSOCD (example), 323
CSRN (theorem), 324
CSROI (example), 325
CSRST (diagram), 326
CSRST (theorem), 327
CSS (theorem), 328
CSSE (subsection, section CRS), 329
CSSOC (subsection, section CRS), 330
CSTW (example), 331
CTD (subsection, section TD), 332
CTLT (example), 333
CUMOS (theorem), 334
curve fitting
    polynomial through 5 points
        example PTFP, 335
CV (definition), 336
CV (notation), 337
CV (technique, section PT), 338
CVA (definition), 339
CVA (notation), 340
CVC (notation), 341
CVE (definition), 342
CVE (notation), 343
CVS (example), 344
CVS (subsection, section VR), 345
CVSM (definition), 346
CVSM (example), 347
CVSM (notation), 348
CVSR (example), 349

D (acronyms, section PDM), 350
D (archetype), 351
D (chapter), 352
D (definition), 353
D (notation), 354
D (section), 355
D (subsection, section D), 356
D (subsection, section SD), 357
D (technique, section PT), 358
D33M (example), 359
DAB (example), 360
DC (example), 361
DC (technique, section PT), 362
DC (theorem), 363
DCM (theorem), 364
DCN (Property), 365
DCP (theorem), 366
DD (subsection, section DM), 367
DEC (theorem), 368
decomposition
    technique DC, 369
DED (theorem), 370
definition
    A, 371
    AM, 372
    AME, 373
    B, 374
    C, 375
    CBM, 376
    CCCV, 377
    CCM, 378
    CCN, 379
    CM, 380
    CNA, 381
    CNE, 382
    CNM, 383
    CP, 384
    CS, 385
    CSM, 386
    CV, 387
    CVA, 388
    CVE, 389
    CVSM, 390
    D, 391
    DIM, 392
    DM, 393
    DS, 394
    DZM, 395
    EEF, 396
    EELT, 397
    EEM, 398
    ELEM, 399
    EM, 400
    EO, 401
    ES, 402
    ESYS, 403
    F, 404
    GES, 405
    GEV, 406
    GME, 407
    HI, 408
    HID, 409
    HM, 410
    HP, 411
    HS, 412
    IDLT, 413
    IDV, 414
    IE, 415
    ILT, 416
    IM, 417
    IMP, 418
    IP, 419
    IS, 420
    IVLT, 421
    IVS, 422
    JB, 423
    JCF, 424
    KLT, 425
    LC, 426
    LCCV, 427
    LI, 428
    LICV, 429
    LNS, 430
    LSS, 431
    LT, 432
    LTA, 433
    LTC, 434
    LTM, 435
    LTR, 436
    LTSM, 437
    M, 438
    MA, 439
    MCN, 440
    ME, 441
    MI, 442
    MM, 443
    MR, 444
    MRLS, 445
    MSM, 446
    MVP, 447
    NLT, 448
    NM, 449
    NOLT, 450
    NOM, 451
    NRML, 452
    NSM, 453
    NV, 454
    ONS, 455
    OSV, 456
    OV, 457
    PI, 458
    PSM, 459
    REM, 460
    RLD, 461
    RLDCV, 462
    RLT, 463
    RO, 464
    ROLT, 465
    ROM, 466
    RR, 467
    RREF, 468
    RSM, 469
    S, 470
    SC, 471
    SE, 472
    SET, 473
    SI, 474
    SIM, 475
    SLE, 476
    SLT, 477
    SM, 478
    SOLV, 479
    SQM, 480
    SRM, 481
    SS, 482
    SSCV, 483
    SSET, 484
    SU, 485
    SUV, 486
    SV, 487
    SYM, 488
    T, 489
    technique D, 490
    TM, 491
    TS, 492
    TSHSE, 493
    TSVS, 494
    UM, 495
    UTM, 496
    VM, 497
    VOC, 498
    VR, 499
    VS, 500
    VSCV, 501
    VSM, 502
    ZCV, 503
    ZM, 504
DEHD (example), 505
DEM (theorem), 506
DEMMM (theorem), 507
DEMS5 (example), 508
DER (theorem), 509
DERC (theorem), 510
determinant
    computed two ways
        example TCSD, 511
    definition DM, 512
    equal rows or columns
        theorem DERC, 513
    expansion, columns
        theorem DEC, 514
    expansion, rows
        theorem DER, 515
    identity matrix
        theorem DIM, 516
    matrix multiplication
        theorem DRMM, 517
    nonsingular matrix, 518
    notation, 519
    row or column multiple
        theorem DRCM, 520
    row or column swap
        theorem DRCS, 521
    size 2 matrix
        theorem DMST, 522
    size 3 matrix
        example D33M, 523
    transpose
        theorem DT, 524
    via row operations
        example DRO, 525
    zero
        theorem SMZD, 526
    zero row or column
        theorem DZRC, 527
    zero versus nonzero
        example ZNDAB, 528
determinant, upper triangular matrix
    example DUTM, 529
determinants
    elementary matrices
        theorem DEMMM, 530
DF (Property), 531
DF (subsection, section CF), 532
DFS (subsection, section PD), 533
DFS (theorem), 534
DGES (theorem), 535
diagonal matrix
    definition DIM, 536
diagonalizable
    definition DZM, 537
    distinct eigenvalues
        example DEHD, 538
        theorem DED, 539
    full eigenspaces
        theorem DMFE, 540
    not
        example NDMS4, 541
diagonalizable matrix
    high power
        example HPDM, 542
diagonalization
    Archetype B
        example DAB, 543
    criteria
        theorem DC, 544
    example DMS3, 545
diagram
    CSRST, 546
    DLTA, 547
    DLTM, 548
    DTSLS, 549
    FTMR, 550
    FTMRA, 551
    GLT, 552
    ILT, 553
    MRCLT, 554
    NILT, 555
DIM (definition), 556
DIM (theorem), 557
dimension
    crazy vector space
        example DC, 558
    definition D, 559
    notation, 560
    polynomial subspace
        example DSP4, 561
    proper subspaces
        theorem PSSD, 562
    subspace
        example DSM22, 563
direct sum
    decomposing zero vector
        theorem DSZV, 564
    definition DS, 565
    dimension
        theorem DSD, 566
    example SDS, 567
    from a basis
        theorem DSFB, 568
    from one subspace
        theorem DSFOS, 569
    notation, 570
    zero intersection
        theorem DSZI, 571
direct sums
    linear independence
        theorem DSLI, 572
    repeated
        theorem RDS, 573
distributivity
    complex numbers
        Property DCN, 574
    field
        Property DF, 575
distributivity, matrix addition
    matrices
        Property DMAM, 576
distributivity, scalar addition
    column vectors
        Property DSAC, 577
    matrices
        Property DSAM, 578
    vectors
        Property DSA, 579
distributivity, vector addition
    column vectors
        Property DVAC, 580
    vectors
        Property DVA, 581
DLDS (theorem), 582
DLTA (diagram), 583
DLTM (diagram), 584
DM (definition), 585
DM (notation), 586
DM (section), 587
DM (theorem), 588
DMAM (Property), 589
DMFE (theorem), 590
DMHP (subsection, section HP), 591
DMHP (theorem), 592
DMMP (theorem), 593
DMS3 (example), 594
DMST (theorem), 595
DNLT (theorem), 596
DNMMM (subsection, section PDM), 597
DP (theorem), 598
DRCM (theorem), 599
DRCMA (theorem), 600
DRCS (theorem), 601
DRMM (theorem), 602
DRO (example), 603
DRO (subsection, section PDM), 604
DROEM (subsection, section PDM), 605
DS (definition), 606
DS (notation), 607
DS (subsection, section PD), 608
DSA (Property), 609
DSAC (Property), 610
DSAM (Property), 611
DSD (theorem), 612
DSFB (theorem), 613
DSFOS (theorem), 614
DSLI (theorem), 615
DSM22 (example), 616
DSP4 (example), 617
DSZI (theorem), 618
DSZV (theorem), 619
DT (theorem), 620
DTSLS (diagram), 621
DUTM (example), 622
DVA (Property), 623
DVAC (Property), 624
DVM (theorem), 625
DVS (subsection, section D), 626
DZM (definition), 627
DZRC (theorem), 628

E (acronyms, section SD), 629
E (archetype), 630
E (chapter), 631
E (technique, section PT), 632
E.SAGE (computation, section SAGE), 633
ECEE (subsection, section EE), 634
EDELI (theorem), 635
EDYES (theorem), 636
EE (section), 637
EEE (subsection, section EE), 638
EEF (definition), 639
EEF (subsection, section FS), 640
EELT (definition), 641
EELT (subsection, section CB), 642
EEM (definition), 643
EEM (subsection, section EE), 644
EEMAP (theorem), 645
EENS (example), 646
EER (theorem), 647
EESR (theorem), 648
EHM (subsection, section PEE), 649
eigenspace
    as null space
        theorem EMNS, 650
    definition EM, 651
    invariant subspace
        theorem EIS, 652
    subspace
        theorem EMS, 653
eigenspaces
    sage, 654
eigenvalue
    algebraic multiplicity
        definition AME, 655
        notation, 656
    complex
        example CEMS6, 657
    definition EEM, 658
    existence
        example CAEHW, 659
        theorem EMHE, 660
    geometric multiplicity
        definition GME, 661
        notation, 662
    index, 663
    linear transformation
        definition EELT, 664
    multiplicities
        example EMMS4, 665
    power
        theorem EOMP, 666
    root of characteristic polynomial
        theorem EMRCP, 667
    scalar multiple
        theorem ESMM, 668
    symmetric matrix
        example ESMS4, 669
    zero
        theorem SMZE, 670
eigenvalues
    building desired
        example BDE, 671
    complex, of a linear transformation
        example CELT, 672
    conjugate pairs
        theorem ERMCP, 673
    distinct
        example DEMS5, 674
    example SEE, 675
    Hermitian matrices
        theorem HMRE, 676
    inverse
        theorem EIM, 677
    maximum number
        theorem MNEM, 678
    multiplicities
        example HMEM5, 679
        theorem ME, 680
    number
        theorem NEM, 681
    of a polynomial
        theorem EPM, 682
    size 3 matrix
        example EMS3, 683
        example ESMS3, 684
    transpose
        theorem ETM, 685
eigenvalues, eigenvectors
    vector, matrix representations
        theorem EER, 686
eigenvector, 687
    linear transformation, 688
eigenvectors, 689
    conjugate pairs, 690
    Hermitian matrices
        theorem HMOE, 691
    linear transformation
        example ELTBM, 692
        example ELTBP, 693
    linearly independent
        theorem EDELI, 694
    of a linear transformation
        example ELTT, 695
EILT (subsection, section ILT), 696
EIM (theorem), 697
EIS (example), 698
EIS (theorem), 699
ELEM (definition), 700
ELEM (notation), 701
elementary matrices
    definition ELEM, 702
    determinants
        theorem DEM, 703
    nonsingular
        theorem EMN, 704
    notation, 705
    row operations
        example EMRO, 706
        theorem EMDRO, 707
ELIS (theorem), 708
ELTBM (example), 709
ELTBP (example), 710
ELTT (example), 711
EM (definition), 712
EM (subsection, section DM), 713
EMDRO (theorem), 714
EMHE (theorem), 715
EMMS4 (example), 716
EMMVP (theorem), 717
EMN (theorem), 718
EMNS (theorem), 719
EMP (theorem), 720
empty set, 721
    notation, 722
EMRCP (theorem), 723
EMRO (example), 724
EMS (theorem), 725
EMS3 (example), 726
ENLT (theorem), 727
EO (definition), 728
EOMP (theorem), 729
EOPSS (theorem), 730
EPM (theorem), 731
EPSM (theorem), 732
equal matrices
    via equal matrix-vector products
        theorem EMMVP, 733
equation operations
    definition EO, 734
    theorem EOPSS, 735
equivalence statements
    technique E, 736
equivalences
    technique ME, 737
equivalent systems
    definition ESYS, 738
ERMCP (theorem), 739
ES (definition), 740
ES (notation), 741
ESEO (subsection, section SSLE), 742
ESLT (subsection, section SLT), 743
ESMM (theorem), 744
ESMS3 (example), 745
ESMS4 (example), 746
ESYS (definition), 747
ETM (theorem), 748
EVS (subsection, section VS), 749
example
    AALC, 750
    ABLC, 751
    ABS, 752
    ACN, 753
    AHSAC, 754
    AIVLT, 755
    ALT, 756
    ALTMM, 757
    AM, 758
    AMAA, 759
    ANILT, 760
    ANM, 761
    AOS, 762
    ASC, 763
    AVR, 764
    BC, 765
    BDE, 766
    BDM22, 767
    BM, 768
    BP, 769
    BPR, 770
    BRLT, 771
    BSM22, 772
    BSP4, 773
    CABAK, 774
    CAEHW, 775
    CBCV, 776
    CBP, 777
    CCM, 778
    CELT, 779
    CEMS6, 780
    CFNLT, 781
    CFV, 782
    CIVLT, 783
    CM32, 784
    CMI, 785
    CMIAB, 786
    CNS1, 787
    CNS2, 788
    CNSV, 789
    COV, 790
    CP2, 791
    CPMS3, 792
    CROB3, 793
    CROB4, 794
    CS, 795
    CSAA, 796
    CSAB, 797
    CSANS, 798
    CSCN, 799
    CSIP, 800
    CSMCS, 801
    CSOCD, 802
    CSROI, 803
    CSTW, 804
    CTLT, 805
    CVS, 806
    CVSM, 807
    CVSR, 808
    D33M, 809
    DAB, 810
    DC, 811
    DEHD, 812
    DEMS5, 813
    DMS3, 814
    DRO, 815
    DSM22, 816
    DSP4, 817
    DUTM, 818
    EENS, 819
    EIS, 820
    ELTBM, 821
    ELTBP, 822
    ELTT, 823
    EMMS4, 824
    EMRO, 825
    EMS3, 826
    ESMS3, 827
    ESMS4, 828
    FDV, 829
    FF8, 830
    FRAN, 831
    FS1, 832
    FS2, 833
    FSAG, 834
    FSCF, 835
    GE4, 836
    GE6, 837
    GENR6, 838
    GSTV, 839
    HISAA, 840
    HISAD, 841
    HMEM5, 842
    HP, 843
    HPDM, 844
    HUSAB, 845
    IAP, 846
    IAR, 847
    IAS, 848
    IAV, 849
    ILTVR, 850
    IM, 851
    IM11, 852
    IS, 853
    ISJB, 854
    ISMR4, 855
    ISMR6, 856
    ISSI, 857
    IVSAV, 858
    JB4, 859
    JCF10, 860
    KPNLT, 861
    KVMR, 862
    LCM, 863
    LDCAA, 864
    LDHS, 865
    LDP4, 866
    LDRN, 867
    LDS, 868
    LIC, 869
    LICAB, 870
    LIHS, 871
    LIM32, 872
    LINSB, 873
    LIP4, 874
    LIS, 875
    LLDS, 876
    LNS, 877
    LTDB1, 878
    LTDB2, 879
    LTDB3, 880
    LTM, 881
    LTPM, 882
    LTPP, 883
    LTRGE, 884
    MA, 885
    MBC, 886
    MCSM, 887
    MFLT, 888
    MI, 889
    MIVS, 890
    MMNC, 891
    MNSLE, 892
    MOLT, 893
    MPMR, 894
    MRBE, 895
    MRCM, 896
    MSCN, 897
    MSM, 898
    MTV, 899
    MWIAA, 900
    NDMS4, 901
    NIAO, 902
    NIAQ, 903
    NIAQR, 904
    NIDAU, 905
    NJB5, 906
    NKAO, 907
    NLT, 908
    NM, 909
    NM62, 910
    NM64, 911
    NM83, 912
    NRREF, 913
    NSAO, 914
    NSAQ, 915
    NSAQR, 916
    NSC2A, 917
    NSC2S, 918
    NSC2Z, 919
    NSDAT, 920
    NSDS, 921
    NSE, 922
    NSEAI, 923
    NSLE, 924
    NSLIL, 925
    NSNM, 926
    NSR, 927
    NSS, 928
    OLTTR, 929
    ONFV, 930
    ONTV, 931
    OSGMD, 932
    OSMC, 933
    PCVS, 934
    PM, 935
    PSHS, 936
    PTFP, 937
    PTM, 938
    PTMEE, 939
    RAO, 940
    RES, 941
    RNM, 942
    RNSM, 943
    ROD2, 944
    ROD4, 945
    RREF, 946
    RREFN, 947
    RRTI, 948
    RS, 949
    RSAI, 950
    RSB, 951
    RSC4, 952
    RSC5, 953
    RSNS, 954
    RSREM, 955
    RVMR, 956
    S, 957
    SAA, 958
    SAB, 959
    SABMI, 960
    SAE, 961
    SAN, 962
    SAR, 963
    SAV, 964
    SC, 965
    SC3, 966
    SCAA, 967
    SCAB, 968
    SCAD, 969
    SDS, 970
    SEE, 971
    SEEF, 972
    SETM, 973
    SI, 974
    SM2Z7, 975
    SM32, 976
    SMLT, 977
    SMS3, 978
    SMS5, 979
    SP4, 980
    SPIAS, 981
    SRR, 982
    SS, 983
    SS6W, 984
    SSC, 985
    SSET, 986
    SSM22, 987
    SSNS, 988
    SSP, 989
    SSP4, 990
    STLT, 991
    STNE, 992
    SU, 993
    SUVOS, 994
    SVP4, 995
    SYM, 996
    TCSD, 997
    TD4, 998
    TDEE6, 999
    TDSSE, 1000
    TIS, 1001
    TIVS, 1002
    TKAP, 1003
    TLC, 1004
    TM, 1005
    TMP, 1006
    TOV, 1007
    TREM, 1008
    TTS, 1009
    UM3, 1010
    UPM, 1011
    US, 1012
    USR, 1013
    VA, 1014
    VESE, 1015
    VFS, 1016
    VFSAD, 1017
    VFSAI, 1018
    VFSAL, 1019
    VM4, 1020
    VRC4, 1021
    VRP2, 1022
    VSCV, 1023
    VSF, 1024
    VSIM5, 1025
    VSIS, 1026
    VSM, 1027
    VSP, 1028
    VSPUD, 1029
    VSS, 1030
    ZNDAB, 1031
EXC (subsection, section B), 1032
EXC (subsection, section CB), 1033
EXC (subsection, section CF), 1034
EXC (subsection, section CRS), 1035
EXC (subsection, section D), 1036
EXC (subsection, section DM), 1037
EXC (subsection, section EE), 1038
EXC (subsection, section F), 1039
EXC (subsection, section FS), 1040
EXC (subsection, section HP), 1041
EXC (subsection, section HSE), 1042
EXC (subsection, section ILT), 1043
EXC (subsection, section IVLT), 1044
EXC (subsection, section LC), 1045
EXC (subsection, section LDS), 1046
EXC (subsection, section LI), 1047
EXC (subsection, section LISS), 1048
EXC (subsection, section LT), 1049
EXC (subsection, section MINM), 1050
EXC (subsection, section MISLE), 1051
EXC (subsection, section MM), 1052
EXC (subsection, section MO), 1053
EXC (subsection, section MR), 1054
EXC (subsection, section NM), 1055
EXC (subsection, section O), 1056
EXC (subsection, section PD), 1057
EXC (subsection, section PDM), 1058
EXC (subsection, section PEE), 1059
EXC (subsection, section PSM), 1060
EXC (subsection, section RREF), 1061
EXC (subsection, section S), 1062
EXC (subsection, section SD), 1063
EXC (subsection, section SLT), 1064
EXC (subsection, section SS), 1065
EXC (subsection, section SSLE), 1066
EXC (subsection, section T), 1067
EXC (subsection, section TSS), 1068
EXC (subsection, section VO), 1069
EXC (subsection, section VR), 1070
EXC (subsection, section VS), 1071
EXC (subsection, section WILA), 1072
extended echelon form
    submatrices
        example SEEF, 1073
extended reduced row-echelon form
    properties
        theorem PEEF, 1074

F (archetype), 1075
F (definition), 1076
F (section), 1077
F (subsection, section F), 1078
FDV (example), 1079
FF (subsection, section F), 1080
FF8 (example), 1081
Fibonacci sequence
    example FSCF, 1082
field
    definition F, 1083
FIMP (theorem), 1084
finite field
    size 8
        example FF8, 1085
four subsets
    example FS1, 1086
    example FS2, 1087
four subspaces
    dimension
        theorem DFS, 1088
FRAN (example), 1089
free variables
    example CFV, 1090
free variables, number
    theorem FVCS, 1091
free, independent variables
    example FDV, 1092
FS (section), 1093
FS (subsection, section FS), 1094
FS (subsection, section SD), 1095
FS (theorem), 1096
FS1 (example), 1097
FS2 (example), 1098
FSAG (example), 1099
FSCF (example), 1100
FTMR (diagram), 1101
FTMR (theorem), 1102
FTMRA (diagram), 1103
FV (subsection, section TSS), 1104
FVCS (theorem), 1105

G (archetype), 1106
G (theorem), 1107
GE4 (example), 1108
GE6 (example), 1109
GEE (subsection, section IS), 1110
GEK (theorem), 1111
generalized eigenspace
    as kernel
        theorem GEK, 1112
    definition GES, 1113
    dimension
        theorem DGES, 1114
    dimension 4 domain
        example GE4, 1115
    dimension 6 domain
        example GE6, 1116
    invariant subspace
        theorem GESIS, 1117
    nilpotent restriction
        theorem RGEN, 1118
    nilpotent restrictions, dimension 6 domain
        example GENR6, 1119
    notation, 1120
generalized eigenspace decomposition
    theorem GESD, 1121
generalized eigenvector
    definition GEV, 1122
GENR6 (example), 1123
GES (definition), 1124
GES (notation), 1125
GESD (subsection, section JCF), 1126
GESD (theorem), 1127
GESIS (theorem), 1128
GEV (definition), 1129
GFDL (appendix), 1130
GLT (diagram), 1131
GME (definition), 1132
GME (notation), 1133
goldilocks
    theorem G, 1134
Gram-Schmidt
    column vectors
        theorem GSP, 1135
    three vectors
        example GSTV, 1136
gram-schmidt
    mathematica, 1137
GS (technique, section PT), 1138
GSP (subsection, section O), 1139
GSP (theorem), 1140
GSP.MMA (computation, section MMA), 1141
GSTV (example), 1142
GT (subsection, section PD), 1143

H (archetype), 1144
Hadamard Identity
    notation, 1145
Hadamard identity
    definition HID, 1146
Hadamard Inverse
    notation, 1147
Hadamard inverse
    definition HI, 1148
Hadamard Product
    Diagonalizable Matrices
        theorem DMHP, 1149
    notation, 1150
Hadamard product
    commutativity
        theorem HPC, 1151
    definition HP, 1152
    diagonal matrices
        theorem DMMP, 1153
    distributivity
        theorem HPDAA, 1154
    example HP, 1155
    identity
        theorem HPHID, 1156
    inverse
        theorem HPHI, 1157
    scalar matrix multiplication
        theorem HPSMM, 1158
hermitian
    definition HM, 1159
Hermitian matrix
    inner product
        theorem HMIP, 1160
HI (definition), 1161
HI (notation), 1162
HID (definition), 1163
HID (notation), 1164
HISAA (example), 1165
HISAD (example), 1166
HM (definition), 1167
HM (subsection, section MM), 1168
HMEM5 (example), 1169
HMIP (theorem), 1170
HMOE (theorem), 1171
HMRE (theorem), 1172
HMVEI (theorem), 1173
homogeneous system
    Archetype C
        example AHSAC, 1174
    consistent
        theorem HSC, 1175
    definition HS, 1176
    infinitely many solutions
        theorem HMVEI, 1177
homogeneous systems
    linear independence, 1178
HP (definition), 1179
HP (example), 1180
HP (notation), 1181
HP (section), 1182
HPC (theorem), 1183
HPDAA (theorem), 1184
HPDM (example), 1185
HPHI (theorem), 1186
HPHID (theorem), 1187
HPSMM (theorem), 1188
HS (definition), 1189
HSC (theorem), 1190
HSE (section), 1191
HUSAB (example), 1192

I (archetype), 1193
I (technique, section PT), 1194
IAP (example), 1195
IAR (example), 1196
IAS (example), 1197
IAV (example), 1198
ICBM (theorem), 1199
ICLT (theorem), 1200
identities
    technique PI, 1201
identity matrix
    determinant, 1202
    example IM, 1203
    notation, 1204
IDLT (definition), 1205
IDV (definition), 1206
IE (definition), 1207
IE (notation), 1208
IFDVS (theorem), 1209
IILT (theorem), 1210
ILT (definition), 1211
ILT (diagram), 1212
ILT (section), 1213
ILTB (theorem), 1214
ILTD (subsection, section ILT), 1215
ILTD (theorem), 1216
ILTIS (theorem), 1217
ILTLI (subsection, section ILT), 1218
ILTLI (theorem), 1219
ILTLT (theorem), 1220
ILTVR (example), 1221
IM (definition), 1222
IM (example), 1223
IM (notation), 1224
IM (subsection, section MISLE), 1225
IM11 (example), 1226
IMILT (theorem), 1227
IMP (definition), 1228
IMR (theorem), 1229
inconsistent linear systems
    theorem ISRN, 1230
independent, dependent variables
    definition IDV, 1231
indesxstring
    example SM2Z7, 1232
    example SSET, 1233
index
    eigenvalue
        definition IE, 1234
        notation, 1235
indexstring
    theorem DRCMA, 1236
    theorem OBUTR, 1237
    theorem UMCOB, 1238
induction
    technique I, 1239
infinite solution set
    example ISSI, 1240
infinite solutions, 3 × 4
    example IS, 1241
injective
    example IAP, 1242
    example IAR, 1243
    not
        example NIAO, 1244
        example NIAQ, 1245
        example NIAQR, 1246
    not, by dimension
        example NIDAU, 1247
    polynomials to matrices
        example IAV, 1248
injective linear transformation
    bases
        theorem ILTB, 1249
injective linear transformations
    dimension
        theorem ILTD, 1250
inner product
    anti-commutative
        theorem IPAC, 1251
    example CSIP, 1252
    norm
        theorem IPN, 1253
    notation, 1254
    positive
        theorem PIP, 1255
    scalar multiplication
        theorem IPSM, 1256
    vector addition
        theorem IPVA, 1257
integers
    mod p
        definition IMP, 1258
    mod p, field
        theorem FIMP, 1259
    mod 11
        example IM11, 1260
interpolating polynomial
    theorem IP, 1261
invariant subspace
    definition IS, 1262
    eigenspace, 1263
    eigenspaces
        example EIS, 1264
    example TIS, 1265
    Jordan block
        example ISJB, 1266
    kernels of powers
        theorem KPIS, 1267
inverse
    composition of linear transformations
        theorem ICLT, 1268
    example CMI, 1269
    example MI, 1270
    notation, 1271
    of a matrix, 1272
invertible linear transformation
    defined by invertible matrix
        theorem IMILT, 1273
invertible linear transformations
    composition
        theorem CIVLT, 1274
    computing
        example CIVLT, 1275
IP (definition), 1276
IP (notation), 1277
IP (subsection, section O), 1278
IP (theorem), 1279
IPAC (theorem), 1280
IPN (theorem), 1281
IPSM (theorem), 1282
IPVA (theorem), 1283
IS (definition), 1284
IS (example), 1285
IS (section), 1286
IS (subsection, section IS), 1287
ISJB (example), 1288
ISMR4 (example), 1289
ISMR6 (example), 1290
isomorphic
    multiple vector spaces
        example MIVS, 1291
    vector spaces
        example IVSAV, 1292
isomorphic vector spaces
    dimension
        theorem IVSED, 1293
    example TIVS, 1294
ISRN (theorem), 1295
ISSI (example), 1296
ITMT (theorem), 1297
IV (subsection, section IVLT), 1298
IVLT (definition), 1299
IVLT (section), 1300
IVLT (subsection, section IVLT), 1301
IVLT (subsection, section MR), 1302
IVS (definition), 1303
IVSAV (example), 1304
IVSED (theorem), 1305

J (archetype), 1306
JB (definition), 1307
JB (notation), 1308
JB4 (example), 1309
JCF (definition), 1310
JCF (section), 1311
JCF (subsection, section JCF), 1312
JCF10 (example), 1313
JCFLT (theorem), 1314
Jordan block
    definition JB, 1315
    nilpotent
        theorem NJB, 1316
    notation, 1317
    size 4
        example JB4, 1318
Jordan canonical form
    definition JCF, 1319
    size 10
        example JCF10, 1320

K (archetype), 1321
kernel
    injective linear transformation
        theorem KILT, 1322
    isomorphic to null space
        theorem KNSI, 1323
    linear transformation
        example NKAO, 1324
    notation, 1325
    of a linear transformation
        definition KLT, 1326
    pre-image, 1327
    subspace
        theorem KLTS, 1328
    trivial
        example TKAP, 1329
    via matrix representation
        example KVMR, 1330
KILT (theorem), 1331
KLT (definition), 1332
KLT (notation), 1333
KLT (subsection, section ILT), 1334
KLTS (theorem), 1335
KNSI (theorem), 1336
KPI (theorem), 1337
KPIS (theorem), 1338
KPLT (theorem), 1339
KPNLT (example), 1340
KPNLT (theorem), 1341
KVMR (example), 1342

L (archetype), 1343
L (technique, section PT), 1344
LA (subsection, section WILA), 1345
LC (definition), 1346
LC (section), 1347
LC (subsection, section LC), 1348
LC (technique, section PT), 1349
LCCV (definition), 1350
LCM (example), 1351
LDCAA (example), 1352
LDHS (example), 1353
LDP4 (example), 1354
LDRN (example), 1355
LDS (example), 1356
LDS (section), 1357
LDSS (subsection, section LDS), 1358
least squares
    minimizes residuals
        theorem LSMR, 1359
least squares solution
    definition LSS, 1360
left null space
    as row space, 1361
    definition LNS, 1362
    example LNS, 1363
    notation, 1364
    subspace
        theorem LNSMS, 1365
lemma
    technique LC, 1366
LI (definition), 1367
LI (section), 1368
LI (subsection, section LISS), 1369
LIC (example), 1370
LICAB (example), 1371
LICV (definition), 1372
LIHS (example), 1373
LIM32 (example), 1374
linear combination
    system of equations
        example ABLC, 1375
    definition LC, 1376
    definition LCCV, 1377
    example TLC, 1378
    linear transformation, 1379
    matrices
        example LCM, 1380
    system of equations
        example AALC, 1381
linear combinations
    solutions to linear systems
        theorem SLSLC, 1382
linear dependence
    more vectors than size
        theorem MVSLD, 1383
linear independence
    definition LI, 1384
    definition LICV, 1385
    homogeneous systems
        theorem LIVHS, 1386
    injective linear transformation
        theorem ILTLI, 1387
    matrices
        example LIM32, 1388
    orthogonal, 1389
    r and n
        theorem LIVRN, 1390
linear solve
    mathematica, 1391
    sage, 1392
linear system
    consistent
        theorem RCLS, 1393
    matrix representation
        definition MRLS, 1394
        notation, 1395
linear systems
    notation
        example MNSLE, 1396
        example NSLE, 1397
linear transformation
    polynomials to polynomials
        example LTPP, 1398
    addition
        definition LTA, 1399
        theorem MLTLT, 1400
        theorem SLTLT, 1401
    as matrix multiplication
        example ALTMM, 1402
    basis of range
        example BRLT, 1403
    checking
        example ALT, 1404
    composition
        definition LTC, 1405
        theorem CLTLT, 1406
    defined by a matrix
        example LTM, 1407
    defined on a basis
        example LTDB1, 1408
        example LTDB2, 1409
        example LTDB3, 1410
        theorem LTDB, 1411
    definition LT, 1412
    identity
        definition IDLT, 1413
    injection
        definition ILT, 1414
    inverse
        theorem ILTLT, 1415
    inverse of inverse
        theorem IILT, 1416
    invertible
        definition IVLT, 1417
        example AIVLT, 1418
    invertible, injective and surjective
        theorem ILTIS, 1419
    Jordan canonical form
        theorem JCFLT, 1420
    kernels of powers
        theorem KPLT, 1421
    linear combination
        theorem LTLC, 1422
    matrix of, 1423
        example MFLT, 1424
        example MOLT, 1425
    not
        example NLT, 1426
    not invertible
        example ANILT, 1427
    notation, 1428
    polynomials to matrices
        example LTPM, 1429
    rank plus nullity
        theorem RPNDD, 1430
    restriction
        definition LTR, 1431
        notation, 1432
    scalar multiple
        example SMLT, 1433
    scalar multiplication
        definition LTSM, 1434
    spanning range
        theorem SSRLT, 1435
    sum
        example STLT, 1436
    surjection
        definition SLT, 1437
    vector space of, 1438
    zero vector
        theorem LTTZZ, 1439
linear transformation inverse
    via matrix representation
        example ILTVR, 1440
linear transformation restriction
    on generalized eigenspace
        example LTRGE, 1441
linear transformations
    compositions
        example CTLT, 1442
    from matrices
        theorem MBLT, 1443
linearly dependent
    r < n
        example LDRN, 1444
    via homogeneous system
        example LDHS, 1445
linearly dependent columns
    Archetype A
        example LDCAA, 1446
linearly dependent set
    example LDS, 1447
    linear combinations within
        theorem DLDS, 1448
    polynomials
        example LDP4, 1449
linearly independent
    crazy vector space
        example LIC, 1450
    extending sets
        theorem ELIS, 1451
    polynomials
        example LIP4, 1452
    via homogeneous system
        example LIHS, 1453
linearly independent columns
    Archetype B
        example LICAB, 1454
linearly independent set
    example LIS, 1455
    example LLDS, 1456
LINM (subsection, section LI), 1457
LINSB (example), 1458
LIP4 (example), 1459
LIS (example), 1460
LISS (section), 1461
LISV (subsection, section LI), 1462
LIVHS (theorem), 1463
LIVRN (theorem), 1464
LLDS (example), 1465
LNS (definition), 1466
LNS (example), 1467
LNS (notation), 1468
LNS (subsection, section FS), 1469
LNSMS (theorem), 1470
lower triangular matrix
    definition LTM, 1471
LS.MMA (computation, section MMA), 1472
LS.SAGE (computation, section SAGE), 1473
LSMR (theorem), 1474
LSS (definition), 1475
LT (acronyms, section IVLT), 1476
LT (chapter), 1477
LT (definition), 1478
LT (notation), 1479
LT (section), 1480
LT (subsection, section LT), 1481
LTA (definition), 1482
LTC (definition), 1483
LTC (subsection, section LT), 1484
LTDB (theorem), 1485
LTDB1 (example), 1486
LTDB2 (example), 1487
LTDB3 (example), 1488
LTLC (subsection, section LT), 1489
LTLC (theorem), 1490
LTM (definition), 1491
LTM (example), 1492
LTPM (example), 1493
LTPP (example), 1494
LTR (definition), 1495
LTR (notation), 1496
LTRGE (example), 1497
LTSM (definition), 1498
LTTZZ (theorem), 1499

M (acronyms, section FS), 1500
M (archetype), 1501
M (chapter), 1502
M (definition), 1503
M (notation), 1504
MA (definition), 1505
MA (example), 1506
MA (notation), 1507
MACN (Property), 1508
MAF (Property), 1509
MAP (subsection, section SVD), 1510
mathematica
    gram-schmidt (computation), 1511
    linear solve (computation), 1512
    matrix entry (computation), 1513
    matrix inverse (computation), 1514
    matrix multiplication (computation), 1515
    null space (computation), 1516
    row reduce (computation), 1517
    transpose of a matrix (computation), 1518
    vector form of solutions (computation), 1519
    vector linear combinations (computation), 1520
mathematical language
    technique L, 1521
matrix
    addition
        definition MA, 1522
        notation, 1523
    augmented
        definition AM, 1524
    column space
        definition CSM, 1525
    complex conjugate
        example CCM, 1526
    definition M, 1527
    equality
        definition ME, 1528
        notation, 1529
    example AM, 1530
    identity
        definition IM, 1531
    inverse
        definition MI, 1532
    nonsingular
        definition NM, 1533
    notation, 1534
    of a linear transformation
        theorem MLTCV, 1535
    product
        example PTM, 1536
        example PTMEE, 1537
    product with vector
        definition MVP, 1538
    rectangular, 1539
    row space
        definition RSM, 1540
    scalar multiplication
        definition MSM, 1541
        notation, 1542
    singular, 1543
    square
        definition SQM, 1544
    submatrices
        example SS, 1545
    submatrix
        definition SM, 1546
    symmetric
        definition SYM, 1547
    transpose
        definition TM, 1548
    unitary
        definition UM, 1549
    unitary is invertible
        theorem UMI, 1550
    zero
        definition ZM, 1551
matrix addition
    example MA, 1552
matrix components
    notation, 1553
matrix entry
    mathematica, 1554
    sage, 1555
    ti83, 1556
    ti86, 1557
matrix inverse
    Archetype B, 1558
    computation
        theorem CINM, 1559
    mathematica, 1560
    nonsingular matrix
        theorem NI, 1561
    of a matrix inverse
        theorem MIMI, 1562
    one-sided
        theorem OSIS, 1563
    product
        theorem SS, 1564
    sage, 1565
    scalar multiple
        theorem MISM, 1566
    size 2 matrices
        theorem TTMI, 1567
    transpose
        theorem MIT, 1568
    uniqueness
        theorem MIU, 1569
matrix multiplication
    adjoints
        theorem MMAD, 1570
    associativity
        theorem MMA, 1571
    complex conjugation
        theorem MMCC, 1572
    definition MM, 1573
    distributivity
        theorem MMDAA, 1574
    entry-by-entry
        theorem EMP, 1575
    identity matrix
        theorem MMIM, 1576
    inner product
        theorem MMIP, 1577
    mathematica, 1578
    noncommutative
        example MMNC, 1579
    scalar matrix multiplication
        theorem MMSMM, 1580
    systems of linear equations
        theorem SLEMM, 1581
    transposes
        theorem MMT, 1582
    zero matrix
        theorem MMZM, 1583
matrix product
    as composition of linear transformations
        example MPMR, 1584
matrix representation
    basis of eigenvectors
        example MRBE, 1585
    composition of linear transformations
        theorem MRCLT, 1586
    definition MR, 1587
    invertible
        theorem IMR, 1588
    multiple of a linear transformation
        theorem MRMLT, 1589
    notation, 1590
    restriction to generalized eigenspace
        theorem MRRGE, 1591
    sum of linear transformations
        theorem MRSLT, 1592
    theorem FTMR, 1593
    upper triangular
        theorem UTMR, 1594
matrix representations
    converting with change-of-basis
        example MRCM, 1595
    example OLTTR, 1596
matrix scalar multiplication
    example MSM, 1597
matrix vector space
    dimension
        theorem DM, 1598
matrix-adjoint product
    eigenvalues, eigenvectors
        theorem EEMAP, 1599
matrix-vector product
    example MTV, 1600
    notation, 1601
MBC (example), 1602
MBLT (theorem), 1603
MC (notation), 1604
MCC (subsection, section MO), 1605
MCCN (Property), 1606
MCF (Property), 1607
MCN (definition), 1608
MCN (subsection, section CNO), 1609
MCSM (example), 1610
MCT (theorem), 1611
MD (chapter), 1612
ME (definition), 1613
ME (notation), 1614
ME (subsection, section PEE), 1615
ME (technique, section PT), 1616
ME (theorem), 1617
ME.MMA (computation, section MMA), 1618
ME.SAGE (computation, section SAGE), 1619
ME.TI83 (computation, section TI83), 1620
ME.TI86 (computation, section TI86), 1621
MEASM (subsection, section MO), 1622
MFLT (example), 1623
MI (definition), 1624
MI (example), 1625
MI (notation), 1626
MI.MMA (computation, section MMA), 1627
MI.SAGE (computation, section SAGE), 1628
MICN (Property), 1629
MIF (Property), 1630
MIMI (theorem), 1631
MINM (section), 1632
MISLE (section), 1633
MISM (theorem), 1634
MIT (theorem), 1635
MIU (theorem), 1636
MIVS (example), 1637
MLT (subsection, section LT), 1638
MLTCV (theorem), 1639
MLTLT (theorem), 1640
MM (definition), 1641
MM (section), 1642
MM (subsection, section MM), 1643
MM.MMA (computation, section MMA), 1644
MMA (section), 1645
MMA (theorem), 1646
MMAD (theorem), 1647
MMCC (theorem), 1648
MMDAA (theorem), 1649
MMEE (subsection, section MM), 1650
MMIM (theorem), 1651
MMIP (theorem), 1652
MMNC (example), 1653
MMSMM (theorem), 1654
MMT (theorem), 1655
MMZM (theorem), 1656
MNEM (theorem), 1657
MNSLE (example), 1658
MO (section), 1659
MOLT (example), 1660
more variables than equations
    example OSGMD, 1661
    theorem CMVEI, 1662
MPMR (example), 1663
MR (definition), 1664
MR (notation), 1665
MR (section), 1666
MRBE (example), 1667
MRCB (theorem), 1668
MRCLT (diagram), 1669
MRCLT (theorem), 1670
MRCM (example), 1671
MRLS (definition), 1672
MRLS (notation), 1673
MRMLT (theorem), 1674
MRRGE (theorem), 1675
MRS (subsection, section CB), 1676
MRSLT (theorem), 1677
MSCN (example), 1678
MSM (definition), 1679
MSM (example), 1680
MSM (notation), 1681
MTV (example), 1682
multiplicative associativity
    complex numbers
        Property MACN, 1683
multiplicative closure
    complex numbers
        Property MCCN, 1684
    field
        Property MCF, 1685
multiplicative commutativity
    complex numbers
        Property CMCN, 1686
multiplicative inverse
    complex numbers
        Property MICN, 1687
MVNSE (subsection, section RREF), 1688
MVP (definition), 1689
MVP (notation), 1690
MVP (subsection, section MM), 1691
MVSLD (theorem), 1692
MWIAA (example), 1693

N (archetype), 1694
N (subsection, section O), 1695
N (technique, section PT), 1696
NDMS4 (example), 1697
negation of statements
    technique N, 1698
NEM (theorem), 1699
NI (theorem), 1700
NIAO (example), 1701
NIAQ (example), 1702
NIAQR (example), 1703
NIDAU (example), 1704
nilpotent
    linear transformation
        definition NLT, 1705
NILT (diagram), 1706
NJB (theorem), 1707
NJB5 (example), 1708
NKAO (example), 1709
NLT (definition), 1710
NLT (example), 1711
NLT (section), 1712
NLT (subsection, section NLT), 1713
NLTFO (subsection, section LT), 1714
NM (definition), 1715
NM (example), 1716
NM (section), 1717
NM (subsection, section NM), 1718
NM (subsection, section OD), 1719
NM62 (example), 1720
NM64 (example), 1721
NM83 (example), 1722
NME1 (theorem), 1723
NME2 (theorem), 1724
NME3 (theorem), 1725
NME4 (theorem), 1726
NME5 (theorem), 1727
NME6 (theorem), 1728
NME7 (theorem), 1729
NME8 (theorem), 1730
NME9 (theorem), 1731
NMI (subsection, section MINM), 1732
NMLIC (theorem), 1733
NMPEM (theorem), 1734
NMRRI (theorem), 1735
NMTNS (theorem), 1736
NMUS (theorem), 1737
NOILT (theorem), 1738
NOLT (definition), 1739
NOLT (notation), 1740
NOM (definition), 1741
NOM (notation), 1742
nonsingular
    columns as basis
        theorem CNMB, 1743
nonsingular matrices
    linearly independent columns
        theorem NMLIC, 1744
nonsingular matrix
    Archetype B
        example NM, 1745
    column space, 1746
    elementary matrices
        theorem NMPEM, 1747
    equivalences
        theorem NME1, 1748
        theorem NME2, 1749
        theorem NME3, 1750
        theorem NME4, 1751
        theorem NME5, 1752
        theorem NME6, 1753
        theorem NME7, 1754
        theorem NME8, 1755
        theorem NME9, 1756
    matrix inverse, 1757
    null space
        example NSNM, 1758
    nullity, 1759
    product of nonsingular matrices
        theorem NPNT, 1760
    rank
        theorem RNNM, 1761
    row-reduced
        theorem NMRRI, 1762
    trivial null space
        theorem NMTNS, 1763
    unique solutions
        theorem NMUS, 1764
nonsingular matrix, row-reduced
    example NSR, 1765
norm
    example CNSV, 1766
    inner product, 1767
    notation, 1768
normal matrix
    definition NRML, 1769
    example ANM, 1770
    orthonormal basis, 1771
notation
    A, 1772
    AM, 1773
    AME, 1774
    C, 1775
    CCCV, 1776
    CCM, 1777
    CCN, 1778
    CNA, 1779
    CNE, 1780
    CNM, 1781
    CSM, 1782
    CV, 1783
    CVA, 1784
    CVC, 1785
    CVE, 1786
    CVSM, 1787
    D, 1788
    DM, 1789
    DS, 1790
    ELEM, 1791
    ES, 1792
    GES, 1793
    GME, 1794
    HI, 1795
    HID, 1796
    HP, 1797
    IE, 1798
    IM, 1799
    IP, 1800
    JB, 1801
    KLT, 1802
    LNS, 1803
    LT, 1804
    LTR, 1805
    M, 1806
    MA, 1807
    MC, 1808
    ME, 1809
    MI, 1810
    MR, 1811
    MRLS, 1812
    MSM, 1813
    MVP, 1814
    NOLT, 1815
    NOM, 1816
    NSM, 1817
    NV, 1818
    RLT, 1819
    RO, 1820
    ROLT, 1821
    ROM, 1822
    RREFA, 1823
    RSM, 1824
    SC, 1825
    SE, 1826
    SETM, 1827
    SI, 1828
    SM, 1829
    SRM, 1830
    SSET, 1831
    SSV, 1832
    SU, 1833
    SUV, 1834
    T, 1835
    TM, 1836
    VR, 1837
    VSCV, 1838
    VSM, 1839
    ZCV, 1840
    ZM, 1841
notation for a linear system
    example NSE, 1842
NPNT (theorem), 1843
NRFO (subsection, section MR), 1844
NRML (definition), 1845
NRREF (example), 1846
NS.MMA (computation, section MMA), 1847
NSAO (example), 1848
NSAQ (example), 1849
NSAQR (example), 1850
NSC2A (example), 1851
NSC2S (example), 1852
NSC2Z (example), 1853
NSDAT (example), 1854
NSDS (example), 1855
NSE (example), 1856
NSEAI (example), 1857
NSLE (example), 1858
NSLIL (example), 1859
NSM (definition), 1860
NSM (notation), 1861
NSM (subsection, section HSE), 1862
NSMS (theorem), 1863
NSNM (example), 1864
NSNM (subsection, section NM), 1865
NSR (example), 1866
NSS (example), 1867
NSSLI (subsection, section LI), 1868
Null space
    as a span
        example NSDS, 1869
null space
    Archetype I
        example NSEAI, 1870
    basis
        theorem BNS, 1871
    computation
        example CNS1, 1872
        example CNS2, 1873
    isomorphic to kernel, 1874
    linearly independent basis
        example LINSB, 1875
    mathematica, 1876
    matrix
        definition NSM, 1877
    nonsingular matrix, 1878
    notation, 1879
    singular matrix, 1880
    spanning set
        example SSNS, 1881
        theorem SSNS, 1882
    subspace
        theorem NSMS, 1883
null space span, linearly independent
    Archetype L
        example NSLIL, 1884
nullity
    computing, 1885
    injective linear transformation
        theorem NOILT, 1886
    linear transformation
        definition NOLT, 1887
    matrix, 1888
        definition NOM, 1889
    notation, 1890, 1891
    square matrix, 1892
NV (definition), 1893
NV (notation), 1894
NVM (theorem), 1895

O (archetype), 1896
O (Property), 1897
O (section), 1898
OBC (subsection, section B), 1899
OBNM (theorem), 1900
OBUTR (theorem), 1901
OC (Property), 1902
OCN (Property), 1903
OD (section), 1904
OD (subsection, section OD), 1905
OD (theorem), 1906
OF (Property), 1907
OLTTR (example), 1908
OM (Property), 1909
one
    column vectors
        Property OC, 1910
    complex numbers
        Property OCN, 1911
    field
        Property OF, 1912
    matrices
        Property OM, 1913
    vectors
        Property O, 1914
ONFV (example), 1915
ONS (definition), 1916
ONTV (example), 1917
orthogonal
    linear independence
        theorem OSLI, 1918
    set
        example AOS, 1919
    set of vectors
        definition OSV, 1920
    vector pairs
        definition OV, 1921
orthogonal vectors
    example TOV, 1922
orthonormal
    definition ONS, 1923
    matrix columns
        example OSMC, 1924
orthonormal basis
    normal matrix
        theorem OBNM, 1925
orthonormal diagonalization
    theorem OD, 1926
orthonormal set
    four vectors
        example ONFV, 1927
    three vectors
        example ONTV, 1928
OSGMD (example), 1929
OSIS (theorem), 1930
OSLI (theorem), 1931
OSMC (example), 1932
OSV (definition), 1933
OV (definition), 1934
OV (subsection, section O), 1935

P (appendix), 1936
P (archetype), 1937
P (technique, section PT), 1938
particular solutions
    example PSHS, 1939
PCNA (theorem), 1940
PCVS (example), 1941
PD (section), 1942
PDM (section), 1943
PDM (theorem), 1944
PEE (section), 1945
PEEF (theorem), 1946
PI (definition), 1947
PI (subsection, section LT), 1948
PI (technique, section PT), 1949
PIP (theorem), 1950
PM (example), 1951
PM (subsection, section EE), 1952
PMI (subsection, section MISLE), 1953
PMM (subsection, section MM), 1954
PMR (subsection, section MR), 1955
PNLT (subsection, section NLT), 1956
POD (section), 1957
polar decomposition
    theorem PDM, 1958
polynomial
    of a matrix
        example PM, 1959
polynomial vector space
    dimension
        theorem DP, 1960
positive semi-definite
    creating
        theorem CPSM, 1961
positive semi-definite matrix
    definition PSM, 1962
    eigenvalues
        theorem EPSM, 1963
practice
    technique P, 1964
pre-image
    definition PI, 1965
    kernel
        theorem KPI, 1966
pre-images
    example SPIAS, 1967
principal axis theorem, 1968
product of triangular matrices
    theorem PTMT, 1969
Property
    AA, 1970
    AAC, 1971
    AACN, 1972
    AAF, 1973
    AAM, 1974
    AC, 1975
    ACC, 1976
    ACCN, 1977
    ACF, 1978
    ACM, 1979
    AI, 1980
    AIC, 1981
    AICN, 1982
    AIF, 1983
    AIM, 1984
    C, 1985
    CACN, 1986
    CAF, 1987
    CC, 1988
    CM, 1989
    CMCN, 1990
    CMF, 1991
    DCN, 1992
    DF, 1993
    DMAM, 1994
    DSA, 1995
    DSAC, 1996
    DSAM, 1997
    DVA, 1998
    DVAC, 1999
    MACN, 2000
    MAF, 2001
    MCCN, 2002
    MCF, 2003
    MICN, 2004
    MIF, 2005
    O, 2006
    OC, 2007
    OCN, 2008
    OF, 2009
    OM, 2010
    SC, 2011
    SCC, 2012
    SCM, 2013
    SMA, 2014
    SMAC, 2015
    SMAM, 2016
    Z, 2017
    ZC, 2018
    ZCN, 2019
    ZF, 2020
    ZM, 2021
PSHS (example), 2022
PSHS (subsection, section LC), 2023
PSM (definition), 2024
PSM (section), 2025
PSM (subsection, section PSM), 2026
PSM (subsection, section SD), 2027
PSMSR (theorem), 2028
PSPHS (theorem), 2029
PSS (subsection, section SSLE), 2030
PSSD (theorem), 2031
PSSLS (theorem), 2032
PT (section), 2033
PTFP (example), 2034
PTM (example), 2035
PTMEE (example), 2036
PTMT (theorem), 2037

Q (archetype), 2038

R (acronyms, section JCF), 2039
R (archetype), 2040
R (chapter), 2041
R.SAGE (computation, section SAGE), 2042
range
    full
        example FRAN, 2043
    isomorphic to column space
        theorem RCSI, 2044
    linear transformation
        example RAO, 2045
    notation, 2046
    of a linear transformation
        definition RLT, 2047
    pre-image
        theorem RPI, 2048
    subspace
        theorem RLTS, 2049
    surjective linear transformation
        theorem RSLT, 2050
    via matrix representation
        example RVMR, 2051
rank
    computing
        theorem CRN, 2052
    linear transformation
        definition ROLT, 2053
    matrix
        definition ROM, 2054
        example RNM, 2055
    notation, 2056, 2057
    of transpose
        example RRTI, 2058
    square matrix
        example RNSM, 2059
    surjective linear transformation
        theorem ROSLT, 2060
    transpose
        theorem RMRT, 2061
rank one decomposition
    size 2
        example ROD2, 2062
    size 4
        example ROD4, 2063
    theorem ROD, 2064
rank+nullity
    theorem RPNC, 2065
RAO (example), 2066
RCLS (theorem), 2067
RCSI (theorem), 2068
RD (subsection, section VS), 2069
RDS (theorem), 2070
READ (subsection, section B), 2071
READ (subsection, section CB), 2072
READ (subsection, section CRS), 2073
READ (subsection, section D), 2074
READ (subsection, section DM), 2075
READ (subsection, section EE), 2076
READ (subsection, section FS), 2077
READ (subsection, section HSE), 2078
READ (subsection, section ILT), 2079
READ (subsection, section IVLT), 2080
READ (subsection, section LC), 2081
READ (subsection, section LDS), 2082
READ (subsection, section LI), 2083
READ (subsection, section LISS), 2084
READ (subsection, section LT), 2085
READ (subsection, section MINM), 2086
READ (subsection, section MISLE), 2087
READ (subsection, section MM), 2088
READ (subsection, section MO), 2089
READ (subsection, section MR), 2090
READ (subsection, section NM), 2091
READ (subsection, section O), 2092
READ (subsection, section PD), 2093
READ (subsection, section PDM), 2094
READ (subsection, section PEE), 2095
READ (subsection, section RREF), 2096
READ (subsection, section S), 2097
READ (subsection, section SD), 2098
READ (subsection, section SLT), 2099
READ (subsection, section SS), 2100
READ (subsection, section SSLE), 2101
READ (subsection, section TSS), 2102
READ (subsection, section VO), 2103
READ (subsection, section VR), 2104
READ (subsection, section VS), 2105
READ (subsection, section WILA), 2106
reduced row-echelon form
    analysis
        notation, 2107
    definition RREF, 2108
    example NRREF, 2109
    example RREF, 2110
    extended
        definition EEF, 2111
    notation
        example RREFN, 2112
    unique
        theorem RREFU, 2113
reducing a span
    example RSC5, 2114
relation of linear dependence
    definition RLD, 2115
    definition RLDCV, 2116
REM (definition), 2117
REMEF (theorem), 2118
REMES (theorem), 2119
REMRS (theorem), 2120
RES (example), 2121
RGEN (theorem), 2122
rings
    sage, 2123
RLD (definition), 2124
RLDCV (definition), 2125
RLT (definition), 2126
RLT (notation), 2127
RLT (subsection, section IS), 2128
RLT (subsection, section SLT), 2129
RLTS (theorem), 2130
RMRT (theorem), 2131
RNLT (subsection, section IVLT), 2132
RNM (example), 2133
RNM (subsection, section D), 2134
RNNM (subsection, section D), 2135
RNNM (theorem), 2136
RNSM (example), 2137
RO (definition), 2138
RO (notation), 2139
RO (subsection, section RREF), 2140
ROD (section), 2141
ROD (theorem), 2142
ROD2 (example), 2143
ROD4 (example), 2144
ROLT (definition), 2145
ROLT (notation), 2146
ROM (definition), 2147
ROM (notation), 2148
ROSLT (theorem), 2149
row operations
    definition RO, 2150
    elementary matrices, 2151, 2152
    notation, 2153
row reduce
    mathematica, 2154
    sage, 2155
    ti83, 2156
    ti86, 2157
row space
    Archetype I
        example RSAI, 2158
    as column space, 2159
    basis
        example RSB, 2160
        theorem BRS, 2161
    matrix, 2162
    notation, 2163
    row-equivalent matrices
        theorem REMRS, 2164
    subspace
        theorem RSMS, 2165
row-equivalent matrices
    definition REM, 2166
    example TREM, 2167
    row space, 2168
    row spaces
        example RSREM, 2169
    theorem REMES, 2170
row-reduce
    the verb
        definition RR, 2171
row-reduced matrices
    theorem REMEF, 2172
RPI (theorem), 2173
RPNC (theorem), 2174
RPNDD (theorem), 2175
RR (definition), 2176
RR.MMA (computation, section MMA), 2177
RR.SAGE (computation, section SAGE), 2178
RR.TI83 (computation, section TI83), 2179
RR.TI86 (computation, section TI86), 2180
RREF (definition), 2181
RREF (example), 2182
RREF (section), 2183
RREF (subsection, section RREF), 2184
RREFA (notation), 2185
RREFN (example), 2186
RREFU (theorem), 2187
RRTI (example), 2188
RS (example), 2189
RSAI (example), 2190
RSB (example), 2191
RSC4 (example), 2192
RSC5 (example), 2193
RSLT (theorem), 2194
RSM (definition), 2195
RSM (notation), 2196
RSM (subsection, section CRS), 2197
RSMS (theorem), 2198
RSNS (example), 2199
RSREM (example), 2200
RT (subsection, section PD), 2201
RVMR (example), 2202

S (archetype), 2203
S (definition), 2204
S (example), 2205
S (section), 2206
SAA (example), 2207
SAB (example), 2208
SABMI (example), 2209
SAE (example), 2210
sage
    eigenspaces (computation), 2211
    linear solve (computation), 2212
    matrix entry (computation), 2213
    matrix inverse (computation), 2214
    rings (computation), 2215
    row reduce (computation), 2216
    transpose of a matrix (computation), 2217
    vector linear combinations (computation), 2218
SAGE (section), 2219
SAN (example), 2220
SAR (example), 2221
SAS (section), 2222
SAV (example), 2223
SC (definition), 2224
SC (example), 2225
SC (notation), 2226
SC (Property), 2227
SC (subsection, section S), 2228
SC (subsection, section SET), 2229
SC3 (example), 2230
SCAA (example), 2231
SCAB (example), 2232
SCAD (example), 2233
scalar closure
    column vectors
        Property SCC, 2234
    matrices
        Property SCM, 2235
    vectors
        Property SC, 2236
scalar multiple
    matrix inverse, 2237
scalar multiplication
    zero scalar
        theorem ZSSM, 2238
    zero vector
        theorem ZVSM, 2239
    zero vector result
        theorem SMEZV, 2240
scalar multiplication associativity
    column vectors
        Property SMAC, 2241
    matrices
        Property SMAM, 2242
    vectors
        Property SMA, 2243
SCB (theorem), 2244
SCC (Property), 2245
SCM (Property), 2246
SD (section), 2247
SDS (example), 2248
SE (definition), 2249
SE (notation), 2250
secret sharing
    6 ways
        example SS6W, 2251
SEE (example), 2252
SEEF (example), 2253
SER (theorem), 2254
set
    cardinality
        definition C, 2255
        example CS, 2256
        notation, 2257
    complement
        definition SC, 2258
        example SC, 2259
        notation, 2260
    definition SET, 2261
    empty
        definition ES, 2262
    equality
        definition SE, 2263
        notation, 2264
    intersection
        definition SI, 2265
        example SI, 2266
        notation, 2267
    membership
        example SETM, 2268
        notation, 2269
    size, 2270
    subset, 2271
    union
        definition SU, 2272
        example SU, 2273
        notation, 2274
SET (definition), 2275
SET (section), 2276
SETM (example), 2277
SETM (notation), 2278
shoes, 2279
SHS (subsection, section HSE), 2280
SI (definition), 2281
SI (example), 2282
SI (notation), 2283
SI (subsection, section IVLT), 2284
SIM (definition), 2285
similar matrices
    equal eigenvalues
        example EENS, 2286
    eual eigenvalues
        theorem SMEE, 2287
    example SMS3, 2288
    example SMS5, 2289
similarity
    definition SIM, 2290
    equivalence relation
        theorem SER, 2291
singular matrix
    Archetype A
        example S, 2292
    null space
        example NSS, 2293
singular matrix, row-reduced
    example SRR, 2294
singular value decomposition
    theorem SVD, 2295
singular values
    definition SV, 2296
SLE (acronyms, section NM), 2297
SLE (chapter), 2298
SLE (definition), 2299
SLE (subsection, section SSLE), 2300
SLELT (subsection, section IVLT), 2301
SLEMM (theorem), 2302
SLSLC (theorem), 2303
SLT (definition), 2304
SLT (section), 2305
SLTB (theorem), 2306
SLTD (subsection, section SLT), 2307
SLTD (theorem), 2308
SLTLT (theorem), 2309
SM (definition), 2310
SM (notation), 2311
SM (subsection, section SD), 2312
SM2Z7 (example), 2313
SM32 (example), 2314
SMA (Property), 2315
SMAC (Property), 2316
SMAM (Property), 2317
SMEE (theorem), 2318
SMEZV (theorem), 2319
SMLT (example), 2320
SMS (theorem), 2321
SMS3 (example), 2322
SMS5 (example), 2323
SMZD (theorem), 2324
SMZE (theorem), 2325
SNCM (theorem), 2326
SO (subsection, section SET), 2327
socks, 2328
SOL (subsection, section B), 2329
SOL (subsection, section CB), 2330
SOL (subsection, section CRS), 2331
SOL (subsection, section D), 2332
SOL (subsection, section DM), 2333
SOL (subsection, section EE), 2334
SOL (subsection, section F), 2335
SOL (subsection, section FS), 2336
SOL (subsection, section HSE), 2337
SOL (subsection, section ILT), 2338
SOL (subsection, section IVLT), 2339
SOL (subsection, section LC), 2340
SOL (subsection, section LDS), 2341
SOL (subsection, section LI), 2342
SOL (subsection, section LISS), 2343
SOL (subsection, section LT), 2344
SOL (subsection, section MINM), 2345
SOL (subsection, section MISLE), 2346
SOL (subsection, section MM), 2347
SOL (subsection, section MO), 2348
SOL (subsection, section MR), 2349
SOL (subsection, section NM), 2350
SOL (subsection, section O), 2351
SOL (subsection, section PD), 2352
SOL (subsection, section PDM), 2353
SOL (subsection, section PEE), 2354
SOL (subsection, section RREF), 2355
SOL (subsection, section S), 2356
SOL (subsection, section SD), 2357
SOL (subsection, section SLT), 2358
SOL (subsection, section SS), 2359
SOL (subsection, section SSLE), 2360
SOL (subsection, section T), 2361
SOL (subsection, section TSS), 2362
SOL (subsection, section VO), 2363
SOL (subsection, section VR), 2364
SOL (subsection, section VS), 2365
SOL (subsection, section WILA), 2366
solution set
    Archetype A
        example SAA, 2367
    archetype E
        example SAE, 2368
    theorem PSPHS, 2369
solution sets
    possibilities
        theorem PSSLS, 2370
solution vector
    definition SOLV, 2371
SOLV (definition), 2372
solving homogeneous system
    Archetype A
        example HISAA, 2373
    Archetype B
        example HUSAB, 2374
    Archetype D
        example HISAD, 2375
solving nonlinear equations
    example STNE, 2376
SP4 (example), 2377
span
    basic
        example ABS, 2378
    basis
        theorem BS, 2379
    definition SS, 2380
    definition SSCV, 2381
    improved
        example IAS, 2382
    notation, 2383
    reducing
        example RSC4, 2384
    reduction
        example RS, 2385
    removing vectors
        example COV, 2386
    reworking elements
        example RES, 2387
    set of polynomials
        example SSP, 2388
    subspace
        theorem SSS, 2389
span of columns
    Archetype A
        example SCAA, 2390
    Archetype B
        example SCAB, 2391
    Archetype D
        example SCAD, 2392
spanning set
    crazy vector space
        example SSC, 2393
    definition TSVS, 2394
    matrices
        example SSM22, 2395
    more vectors
        theorem SSLD, 2396
    polynomials
        example SSP4, 2397
SPIAS (example), 2398
SQM (definition), 2399
square root
    eigenvalues, eigenspaces
        theorem EESR, 2400
    matrix
        definition SRM, 2401
        notation, 2402
    positive semi-definite matrix
        theorem PSMSR, 2403
    unique
        theorem USR, 2404
SR (section), 2405
SRM (definition), 2406
SRM (notation), 2407
SRM (subsection, section SR), 2408
SRR (example), 2409
SS (definition), 2410
SS (example), 2411
SS (section), 2412
SS (subsection, section LISS), 2413
SS (theorem), 2414
SS6W (example), 2415
SSC (example), 2416
SSCV (definition), 2417
SSET (definition), 2418
SSET (example), 2419
SSET (notation), 2420
SSLD (theorem), 2421
SSLE (section), 2422
SSM22 (example), 2423
SSNS (example), 2424
SSNS (subsection, section SS), 2425
SSNS (theorem), 2426
SSP (example), 2427
SSP4 (example), 2428
SSRLT (theorem), 2429
SSS (theorem), 2430
SSSLT (subsection, section SLT), 2431
SSV (notation), 2432
SSV (subsection, section SS), 2433
standard unit vector
    notation, 2434
starting proofs
    technique GS, 2435
STLT (example), 2436
STNE (example), 2437
SU (definition), 2438
SU (example), 2439
SU (notation), 2440
submatrix
    notation, 2441
subset
    definition SSET, 2442
    notation, 2443
subspace
    as null space
        example RSNS, 2444
    characterized
        example ASC, 2445
    definition S, 2446
    in {P}_{4}
        example SP4, 2447
    not, additive closure
        example NSC2A, 2448
    not, scalar closure
        example NSC2S, 2449
    not, zero vector
        example NSC2Z, 2450
    testing
        theorem TSS, 2451
    trivial
        definition TS, 2452
    verification
        example SC3, 2453
        example SM32, 2454
subspaces
    equal dimension
        theorem EDYES, 2455
surjective
    Archetype N
        example SAN, 2456
    example SAR, 2457
    not
        example NSAQ, 2458
        example NSAQR, 2459
    not, Archetype O
        example NSAO, 2460
    not, by dimension
        example NSDAT, 2461
    polynomials to matrices
        example SAV, 2462
surjective linear transformation
    bases
        theorem SLTB, 2463
surjective linear transformations
    dimension
        theorem SLTD, 2464
SUV (definition), 2465
SUV (notation), 2466
SUVB (theorem), 2467
SUVOS (example), 2468
SV (definition), 2469
SVD (section), 2470
SVD (subsection, section SVD), 2471
SVD (theorem), 2472
SVP4 (example), 2473
SYM (definition), 2474
SYM (example), 2475
symmetric matrices
    theorem SMS, 2476
symmetric matrix
    example SYM, 2477
system of equations
    vector equality
        example VESE, 2478
system of linear equations
    definition SLE, 2479

T (archetype), 2480
T (definition), 2481
T (notation), 2482
T (part), 2483
T (section), 2484
T (technique, section PT), 2485
TCSD (example), 2486
TD (section), 2487
TD (subsection, section TD), 2488
TD (theorem), 2489
TD4 (example), 2490
TDEE (theorem), 2491
TDEE6 (example), 2492
TDSSE (example), 2493
TDSSE (subsection, section TD), 2494
technique
    C, 2495
    CD, 2496
    CP, 2497
    CV, 2498
    D, 2499
    DC, 2500
    E, 2501
    GS, 2502
    I, 2503
    L, 2504
    LC, 2505
    ME, 2506
    N, 2507
    P, 2508
    PI, 2509
    T, 2510
    U, 2511
theorem
    AA, 2512
    AIP, 2513
    AISM, 2514
    AIU, 2515
    AMA, 2516
    AMSM, 2517
    BCS, 2518
    BIS, 2519
    BNS, 2520
    BRS, 2521
    BS, 2522
    CB, 2523
    CCM, 2524
    CCRA, 2525
    CCRM, 2526
    CCT, 2527
    CFDVS, 2528
    CFNLT, 2529
    CHT, 2530
    CILTI, 2531
    CINM, 2532
    CIVLT, 2533
    CLI, 2534
    CLTLT, 2535
    CMVEI, 2536
    CNMB, 2537
    COB, 2538
    CPSM, 2539
    CRMA, 2540
    CRMSM, 2541
    CRN, 2542
    CRSM, 2543
    CRVA, 2544
    CSCS, 2545
    CSLTS, 2546
    CSMS, 2547
    CSNM, 2548
    CSRN, 2549
    CSRST, 2550
    CSS, 2551
    CUMOS, 2552
    DC, 2553
    DCM, 2554
    DCP, 2555
    DEC, 2556
    DED, 2557
    DEM, 2558
    DEMMM, 2559
    DER, 2560
    DERC, 2561
    DFS, 2562
    DGES, 2563
    DIM, 2564
    DLDS, 2565
    DM, 2566
    DMFE, 2567
    DMHP, 2568
    DMMP, 2569
    DMST, 2570
    DNLT, 2571
    DP, 2572
    DRCM, 2573
    DRCMA, 2574
    DRCS, 2575
    DRMM, 2576
    DSD, 2577
    DSFB, 2578
    DSFOS, 2579
    DSLI, 2580
    DSZI, 2581
    DSZV, 2582
    DT, 2583
    DVM, 2584
    DZRC, 2585
    EDELI, 2586
    EDYES, 2587
    EEMAP, 2588
    EER, 2589
    EESR, 2590
    EIM, 2591
    EIS, 2592
    ELIS, 2593
    EMDRO, 2594
    EMHE, 2595
    EMMVP, 2596
    EMN, 2597
    EMNS, 2598
    EMP, 2599
    EMRCP, 2600
    EMS, 2601
    ENLT, 2602
    EOMP, 2603
    EOPSS, 2604
    EPM, 2605
    EPSM, 2606
    ERMCP, 2607
    ESMM, 2608
    ETM, 2609
    FIMP, 2610
    FS, 2611
    FTMR, 2612
    FVCS, 2613
    G, 2614
    GEK, 2615
    GESD, 2616
    GESIS, 2617
    GSP, 2618
    HMIP, 2619
    HMOE, 2620
    HMRE, 2621
    HMVEI, 2622
    HPC, 2623
    HPDAA, 2624
    HPHI, 2625
    HPHID, 2626
    HPSMM, 2627
    HSC, 2628
    ICBM, 2629
    ICLT, 2630
    IFDVS, 2631
    IILT, 2632
    ILTB, 2633
    ILTD, 2634
    ILTIS, 2635
    ILTLI, 2636
    ILTLT, 2637
    IMILT, 2638
    IMR, 2639
    IP, 2640
    IPAC, 2641
    IPN, 2642
    IPSM, 2643
    IPVA, 2644
    ISRN, 2645
    ITMT, 2646
    IVSED, 2647
    JCFLT, 2648
    KILT, 2649
    KLTS, 2650
    KNSI, 2651
    KPI, 2652
    KPIS, 2653
    KPLT, 2654
    KPNLT, 2655
    LIVHS, 2656
    LIVRN, 2657
    LNSMS, 2658
    LSMR, 2659
    LTDB, 2660
    LTLC, 2661
    LTTZZ, 2662
    MBLT, 2663
    MCT, 2664
    ME, 2665
    MIMI, 2666
    MISM, 2667
    MIT, 2668
    MIU, 2669
    MLTCV, 2670
    MLTLT, 2671
    MMA, 2672
    MMAD, 2673
    MMCC, 2674
    MMDAA, 2675
    MMIM, 2676
    MMIP, 2677
    MMSMM, 2678
    MMT, 2679
    MMZM, 2680
    MNEM, 2681
    MRCB, 2682
    MRCLT, 2683
    MRMLT, 2684
    MRRGE, 2685
    MRSLT, 2686
    MVSLD, 2687
    NEM, 2688
    NI, 2689
    NJB, 2690
    NME1, 2691
    NME2, 2692
    NME3, 2693
    NME4, 2694
    NME5, 2695
    NME6, 2696
    NME7, 2697
    NME8, 2698
    NME9, 2699
    NMLIC, 2700
    NMPEM, 2701
    NMRRI, 2702
    NMTNS, 2703
    NMUS, 2704
    NOILT, 2705
    NPNT, 2706
    NSMS, 2707
    NVM, 2708
    OBNM, 2709
    OBUTR, 2710
    OD, 2711
    OSIS, 2712
    OSLI, 2713
    PCNA, 2714
    PDM, 2715
    PEEF, 2716
    PIP, 2717
    PSMSR, 2718
    PSPHS, 2719
    PSSD, 2720
    PSSLS, 2721
    PTMT, 2722
    RCLS, 2723
    RCSI, 2724
    RDS, 2725
    REMEF, 2726
    REMES, 2727
    REMRS, 2728
    RGEN, 2729
    RLTS, 2730
    RMRT, 2731
    RNNM, 2732
    ROD, 2733
    ROSLT, 2734
    RPI, 2735
    RPNC, 2736
    RPNDD, 2737
    RREFU, 2738
    RSLT, 2739
    RSMS, 2740
    SCB, 2741
    SER, 2742
    SLEMM, 2743
    SLSLC, 2744
    SLTB, 2745
    SLTD, 2746
    SLTLT, 2747
    SMEE, 2748
    SMEZV, 2749
    SMS, 2750
    SMZD, 2751
    SMZE, 2752
    SNCM, 2753
    SS, 2754
    SSLD, 2755
    SSNS, 2756
    SSRLT, 2757
    SSS, 2758
    SUVB, 2759
    SVD, 2760
    TD, 2761
    TDEE, 2762
    technique T, 2763
    TIST, 2764
    TL, 2765
    TMA, 2766
    TMSM, 2767
    TSE, 2768
    TSRM, 2769
    TSS, 2770
    TT, 2771
    TTMI, 2772
    UMCOB, 2773
    UMI, 2774
    UMPIP, 2775
    USR, 2776
    UTMR, 2777
    VFSLS, 2778
    VRI, 2779
    VRILT, 2780
    VRLT, 2781
    VRRB, 2782
    VRS, 2783
    VSLT, 2784
    VSPCV, 2785
    VSPM, 2786
    ZSSM, 2787
    ZVSM, 2788
    ZVU, 2789
ti83
    matrix entry (computation), 2790
    row reduce (computation), 2791
    vector linear combinations (computation), 2792
TI83 (section), 2793
ti86
    matrix entry (computation), 2794
    row reduce (computation), 2795
    transpose of a matrix (computation), 2796
    vector linear combinations (computation), 2797
TI86 (section), 2798
TIS (example), 2799
TIST (theorem), 2800
TIVS (example), 2801
TKAP (example), 2802
TL (theorem), 2803
TLC (example), 2804
TM (definition), 2805
TM (example), 2806
TM (notation), 2807
TM (subsection, section OD), 2808
TM.MMA (computation, section MMA), 2809
TM.SAGE (computation, section SAGE), 2810
TM.TI86 (computation, section TI86), 2811
TMA (theorem), 2812
TMP (example), 2813
TMSM (theorem), 2814
TOV (example), 2815
trace
    definition T, 2816
    linearity
        theorem TL, 2817
    matrix multiplication
        theorem TSRM, 2818
    notation, 2819
    similarity
        theorem TIST, 2820
    sum of eigenvalues
        theorem TSE, 2821
trail mix
    example TMP, 2822
transpose
    matrix scalar multiplication
        theorem TMSM, 2823
    example TM, 2824
    matrix addition
        theorem TMA, 2825
    matrix inverse, 2826, 2827
    notation, 2828
    scalar multiplication, 2829
transpose of a matrix
    mathematica, 2830
    sage, 2831
    ti86, 2832
transpose of a transpose
    theorem TT, 2833
TREM (example), 2834
triangular decomposition
    entry by entry, size 6
        example TDEE6, 2835
    entry by entry
        theorem TDEE, 2836
    size 4
        example TD4, 2837
    solving systems of equations
        example TDSSE, 2838
    theorem TD, 2839
triangular matrix
    inverse
        theorem ITMT, 2840
trivial solution
    system of equations
        definition TSHSE, 2841
TS (definition), 2842
TS (subsection, section S), 2843
TSE (theorem), 2844
TSHSE (definition), 2845
TSM (subsection, section MO), 2846
TSRM (theorem), 2847
TSS (section), 2848
TSS (subsection, section S), 2849
TSS (theorem), 2850
TSVS (definition), 2851
TT (theorem), 2852
TTMI (theorem), 2853
TTS (example), 2854
typical systems, 2 × 2
    example TTS, 2855

U (archetype), 2856
U (technique, section PT), 2857
UM (definition), 2858
UM (subsection, section MINM), 2859
UM3 (example), 2860
UMCOB (theorem), 2861
UMI (theorem), 2862
UMPIP (theorem), 2863
unique solution, 3 × 3
    example US, 2864
    example USR, 2865
uniqueness
    technique U, 2866
unit vectors
    basis
        theorem SUVB, 2867
    definition SUV, 2868
    orthogonal
        example SUVOS, 2869
unitary
    permutation matrix
        example UPM, 2870
    size 3
        example UM3, 2871
unitary matrices
    columns
        theorem CUMOS, 2872
unitary matrix
    inner product
        theorem UMPIP, 2873
UPM (example), 2874
upper triangular matrix
    definition UTM, 2875
US (example), 2876
USR (example), 2877
USR (theorem), 2878
UTM (definition), 2879
UTMR (subsection, section OD), 2880
UTMR (theorem), 2881

V (acronyms, section O), 2882
V (archetype), 2883
V (chapter), 2884
VA (example), 2885
Vandermonde matrix
    definition VM, 2886
vandermonde matrix
    determinant
        theorem DVM, 2887
    nonsingular
        theorem NVM, 2888
    size 4
        example VM4, 2889
VEASM (subsection, section VO), 2890
vector
    addition
        definition CVA, 2891
    column
        definition CV, 2892
    equality
        definition CVE, 2893
        notation, 2894
    inner product
        definition IP, 2895
    norm
        definition NV, 2896
    notation, 2897
    of constants
        definition VOC, 2898
    product with matrix, 2899, 2900
    scalar multiplication
        definition CVSM, 2901
vector addition
    example VA, 2902
vector component
    notation, 2903
vector form of solutions
    Archetype D
        example VFSAD, 2904
    Archetype I
        example VFSAI, 2905
    Archetype L
        example VFSAL, 2906
    example VFS, 2907
    mathematica, 2908
    theorem VFSLS, 2909
vector linear combinations
    mathematica, 2910
    sage, 2911
    ti83, 2912
    ti86, 2913
vector representation
    example AVR, 2914
    example VRC4, 2915
    injective
        theorem VRI, 2916
    invertible
        theorem VRILT, 2917
    linear transformation
        definition VR, 2918
        notation, 2919
        theorem VRLT, 2920
    surjective
        theorem VRS, 2921
    theorem VRRB, 2922
vector representations
    polynomials
        example VRP2, 2923
vector scalar multiplication
    example CVSM, 2924
vector space
    characterization
        theorem CFDVS, 2925
    column vectors
        definition VSCV, 2926
    definition VS, 2927
    infinite dimension
        example VSPUD, 2928
    linear transformations
        theorem VSLT, 2929
    over integers mod 5
        example VSIM5, 2930
vector space of column vectors
    notation, 2931
vector space of functions
    example VSF, 2932
vector space of infinite sequences
    example VSIS, 2933
vector space of matrices
    definition VSM, 2934
    example VSM, 2935
    notation, 2936
vector space of polynomials
    example VSP, 2937
vector space properties
    column vectors
        theorem VSPCV, 2938
    matrices
        theorem VSPM, 2939
vector space, crazy
    example CVS, 2940
vector space, singleton
    example VSS, 2941
vector spaces
    isomorphic
        definition IVS, 2942
        theorem IFDVS, 2943
VESE (example), 2944
VFS (example), 2945
VFSAD (example), 2946
VFSAI (example), 2947
VFSAL (example), 2948
VFSLS (theorem), 2949
VFSS (subsection, section LC), 2950
VFSS.MMA (computation, section MMA), 2951
VLC.MMA (computation, section MMA), 2952
VLC.SAGE (computation, section SAGE), 2953
VLC.TI83 (computation, section TI83), 2954
VLC.TI86 (computation, section TI86), 2955
VM (definition), 2956
VM (section), 2957
VM4 (example), 2958
VO (section), 2959
VOC (definition), 2960
VR (definition), 2961
VR (notation), 2962
VR (section), 2963
VR (subsection, section LISS), 2964
VRC4 (example), 2965
VRI (theorem), 2966
VRILT (theorem), 2967
VRLT (theorem), 2968
VRP2 (example), 2969
VRRB (theorem), 2970
VRS (theorem), 2971
VS (acronyms, section PD), 2972
VS (chapter), 2973
VS (definition), 2974
VS (section), 2975
VS (subsection, section VS), 2976
VSCV (definition), 2977
VSCV (example), 2978
VSCV (notation), 2979
VSF (example), 2980
VSIM5 (example), 2981
VSIS (example), 2982
VSLT (theorem), 2983
VSM (definition), 2984
VSM (example), 2985
VSM (notation), 2986
VSP (example), 2987
VSP (subsection, section MO), 2988
VSP (subsection, section VO), 2989
VSP (subsection, section VS), 2990
VSPCV (theorem), 2991
VSPM (theorem), 2992
VSPUD (example), 2993
VSS (example), 2994

W (archetype), 2995
WILA (section), 2996

X (archetype), 2997

Z (Property), 2998
ZC (Property), 2999
ZCN (Property), 3000
ZCV (definition), 3001
ZCV (notation), 3002
zero
    complex numbers
        Property ZCN, 3003
    field
        Property ZF, 3004
zero column vector
    definition ZCV, 3005
    notation, 3006
zero matrix
    notation, 3007
zero vector
    column vectors
        Property ZC, 3008
    matrices
        Property ZM, 3009
    unique
        theorem ZVU, 3010
    vectors
        Property Z, 3011
ZF (Property), 3012
ZM (definition), 3013
ZM (notation), 3014
ZM (Property), 3015
ZNDAB (example), 3016
ZSSM (theorem), 3017
ZVSM (theorem), 3018
ZVU (theorem), 3019