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
CCS (subsection, section FS), 162
CCT (theorem), 163
CD (subsection, section DM), 164
CD (technique, section PT), 165
CEE (subsection, section EE), 166
CELT (example), 167
CELT (subsection, section CB), 168
CEMS6 (example), 169
CF (section), 170
CFDVS (theorem), 171
CFNLT (example), 172
CFNLT (subsection, section NLT), 173
CFNLT (theorem), 174
CFV (example), 175
change of basis
    between polynomials
        example CBP, 176
change-of-basis
    between column vectors
        example CBCV, 177
    matrix representation
        theorem MRCB, 178
    similarity
        theorem SCB, 179
    theorem CB, 180
change-of-basis matrix
    definition CBM, 181
    inverse
        theorem ICBM, 182
characteristic polynomial
    definition CP, 183
    degree
        theorem DCP, 184
    size 3 matrix
        example CPMS3, 185
CHT (subsection, section JCF), 186
CHT (theorem), 187
CILT (subsection, section ILT), 188
CILTI (theorem), 189
CIM (subsection, section MISLE), 190
CINM (theorem), 191
CIVLT (example), 192
CIVLT (theorem), 193
CLI (theorem), 194
CLTLT (theorem), 195
CM (definition), 196
CM (Property), 197
CM32 (example), 198
CMCN (Property), 199
CMF (Property), 200
CMI (example), 201
CMIAB (example), 202
CMVEI (theorem), 203
CN (appendix), 204
CNA (definition), 205
CNA (notation), 206
CNA (subsection, section CNO), 207
CNE (definition), 208
CNE (notation), 209
CNM (definition), 210
CNM (notation), 211
CNMB (theorem), 212
CNO (section), 213
CNS1 (example), 214
CNS2 (example), 215
CNSV (example), 216
COB (theorem), 217
coefficient matrix
    definition CM, 218
    nonsingular
        theorem SNCM, 219
column space
    as null space
        theorem FS, 220
    Archetype A
        example CSAA, 221
    Archetype B
        example CSAB, 222
    as null space
        example CSANS, 223
    as null space, Archetype G
        example FSAG, 224
    as row space
        theorem CSRST, 225
    basis
        theorem BCS, 226
    consistent system
        theorem CSCS, 227
    consistent systems
        example CSMCS, 228
    isomorphic to range, 229
    matrix, 230
    nonsingular matrix
        theorem CSNM, 231
    notation, 232
    original columns, Archetype D
        example CSOCD, 233
    row operations, Archetype I
        example CSROI, 234
    subspace
        theorem CSMS, 235
    testing membership
        example MCSM, 236
    two computations
        example CSTW, 237
column vector addition
    notation, 238
column vector scalar multiplication
    notation, 239
commutativity
    column vectors
        Property CC, 240
    matrices
        Property CM, 241
    vectors
        Property C, 242
complex m-space
    example VSCV, 243
complex arithmetic
    example ACN, 244
complex number
    conjugate
        example CSCN, 245
    modulus
        example MSCN, 246
complex number
    conjugate
        definition CCN, 247
    modulus
        definition MCN, 248
complex numbers
    addition
        definition CNA, 249
        notation, 250
    arithmetic properties
        theorem PCNA, 251
    equality
        definition CNE, 252
        notation, 253
    multiplication
        definition CNM, 254
        notation, 255
complex vector space
    dimension
        theorem DCM, 256
composition
    injective linear transformations
        theorem CILTI, 257
    surjective linear transformations
        theorem CSLTS, 258
conjugate
    addition
        theorem CCRA, 259
    column vector
        definition CCCV, 260
    matrix
        definition CCM, 261
        notation, 262
    multiplication
        theorem CCRM, 263
    notation, 264
    of conjugate of a matrix
        theorem CCM, 265
    scalar multiplication
        theorem CRSM, 266
    twice
        theorem CCT, 267
    vector addition
        theorem CRVA, 268
conjugate of a vector
    notation, 269
conjugation
    matrix addition
        theorem CRMA, 270
    matrix scalar multiplication
        theorem CRMSM, 271
    matrix transpose
        theorem MCT, 272
consistent linear system, 273
consistent linear systems
    theorem CSRN, 274
consistent system
    definition CS, 275
constructive proofs
    technique C, 276
contradiction
    technique CD, 277
contrapositive
    technique CP, 278
converse
    technique CV, 279
coordinates
    orthonormal basis
        theorem COB, 280
coordinatization
    linear combination of matrices
        example CM32, 281
    linear independence
        theorem CLI, 282
    orthonormal basis
        example CROB3, 283
        example CROB4, 284
    spanning sets
        theorem CSS, 285
coordinatization principle, 286
coordinatizing
    polynomials
        example CP2, 287
COV (example), 288
COV (subsection, section LDS), 289
CP (definition), 290
CP (subsection, section VR), 291
CP (technique, section PT), 292
CP2 (example), 293
CPMS3 (example), 294
CPSM (theorem), 295
crazy vector space
    example CVSR, 296
    properties
        example PCVS, 297
CRMA (theorem), 298
CRMSM (theorem), 299
CRN (theorem), 300
CROB3 (example), 301
CROB4 (example), 302
CRS (section), 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 (archetype), 350
D (chapter), 351
D (definition), 352
D (notation), 353
D (section), 354
D (subsection, section D), 355
D (subsection, section SD), 356
D (technique, section PT), 357
D33M (example), 358
DAB (example), 359
DC (example), 360
DC (technique, section PT), 361
DC (theorem), 362
DCM (theorem), 363
DCN (Property), 364
DCP (theorem), 365
DD (subsection, section DM), 366
DEC (theorem), 367
decomposition
    technique DC, 368
DED (theorem), 369
definition
    A, 370
    AM, 371
    AME, 372
    B, 373
    C, 374
    CBM, 375
    CCCV, 376
    CCM, 377
    CCN, 378
    CM, 379
    CNA, 380
    CNE, 381
    CNM, 382
    CP, 383
    CS, 384
    CSM, 385
    CV, 386
    CVA, 387
    CVE, 388
    CVSM, 389
    D, 390
    DIM, 391
    DM, 392
    DS, 393
    DZM, 394
    EEF, 395
    EELT, 396
    EEM, 397
    ELEM, 398
    EM, 399
    EO, 400
    ES, 401
    ESYS, 402
    F, 403
    GES, 404
    GEV, 405
    GME, 406
    HI, 407
    HID, 408
    HM, 409
    HP, 410
    HS, 411
    IDLT, 412
    IDV, 413
    IE, 414
    ILT, 415
    IM, 416
    IMP, 417
    IP, 418
    IS, 419
    IVLT, 420
    IVS, 421
    JB, 422
    JCF, 423
    KLT, 424
    LC, 425
    LCCV, 426
    LI, 427
    LICV, 428
    LNS, 429
    LSS, 430
    LT, 431
    LTA, 432
    LTC, 433
    LTM, 434
    LTR, 435
    LTSM, 436
    M, 437
    MA, 438
    MCN, 439
    ME, 440
    MI, 441
    MM, 442
    MR, 443
    MRLS, 444
    MSM, 445
    MVP, 446
    NLT, 447
    NM, 448
    NOLT, 449
    NOM, 450
    NRML, 451
    NSM, 452
    NV, 453
    ONS, 454
    OSV, 455
    OV, 456
    PI, 457
    PSM, 458
    REM, 459
    RLD, 460
    RLDCV, 461
    RLT, 462
    RO, 463
    ROLT, 464
    ROM, 465
    RR, 466
    RREF, 467
    RSM, 468
    S, 469
    SC, 470
    SE, 471
    SET, 472
    SI, 473
    SIM, 474
    SLE, 475
    SLT, 476
    SM, 477
    SOLV, 478
    SQM, 479
    SRM, 480
    SS, 481
    SSCV, 482
    SSET, 483
    SSLE, 484
    SSSLE, 485
    SU, 486
    SUV, 487
    SV, 488
    SYM, 489
    T, 490
    technique D, 491
    TM, 492
    TS, 493
    TSHSE, 494
    TSVS, 495
    UM, 496
    UTM, 497
    VM, 498
    VOC, 499
    VR, 500
    VS, 501
    VSCV, 502
    VSM, 503
    ZCV, 504
    ZM, 505
DEHD (example), 506
DEM (theorem), 507
DEMMM (theorem), 508
DEMS5 (example), 509
DER (theorem), 510
DERC (theorem), 511
determinant
    computed two ways
        example TCSD, 512
    definition DM, 513
    equal rows or columns
        theorem DERC, 514
    expansion, columns
        theorem DEC, 515
    expansion, rows
        theorem DER, 516
    identity matrix
        theorem DIM, 517
    matrix multiplication
        theorem DRMM, 518
    nonsingular matrix, 519
    notation, 520
    row or column multiple
        theorem DRCM, 521
    row or column swap
        theorem DRCS, 522
    size 2 matrix
        theorem DMST, 523
    size 3 matrix
        example D33M, 524
    transpose
        theorem DT, 525
    via row operations
        example DRO, 526
    zero
        theorem SMZD, 527
    zero row or column
        theorem DZRC, 528
    zero versus nonzero
        example ZNDAB, 529
determinant, upper triangular matrix
    example DUTM, 530
determinants
    elementary matrices
        theorem DEMMM, 531
DF (Property), 532
DF (subsection, section CF), 533
DFS (subsection, section PD), 534
DFS (theorem), 535
DGES (theorem), 536
diagonal matrix
    definition DIM, 537
diagonalizable
    definition DZM, 538
    distinct eigenvalues
        example DEHD, 539
        theorem DED, 540
    full eigenspaces
        theorem DMFE, 541
    not
        example NDMS4, 542
diagonalizable matrix
    high power
        example HPDM, 543
diagonalization
    Archetype B
        example DAB, 544
    criteria
        theorem DC, 545
    example DMS3, 546
diagram
    CSRST, 547
    DLTA, 548
    DLTM, 549
    DTSLS, 550
    FTMR, 551
    FTMRA, 552
    GLT, 553
    ILT, 554
    MRCLT, 555
    NILT, 556
DIM (definition), 557
DIM (theorem), 558
dimension
    crazy vector space
        example DC, 559
    definition D, 560
    notation, 561
    polynomial subspace
        example DSP4, 562
    proper subspaces
        theorem PSSD, 563
    subspace
        example DSM22, 564
direct sum
    decomposing zero vector
        theorem DSZV, 565
    definition DS, 566
    dimension
        theorem DSD, 567
    example SDS, 568
    from a basis
        theorem DSFB, 569
    from one subspace
        theorem DSFOS, 570
    notation, 571
    zero intersection
        theorem DSZI, 572
direct sums
    linear independence
        theorem DSLI, 573
    repeated
        theorem RDS, 574
distributivity
    complex numbers
        Property DCN, 575
    field
        Property DF, 576
distributivity, matrix addition
    matrices
        Property DMAM, 577
distributivity, scalar addition
    column vectors
        Property DSAC, 578
    matrices
        Property DSAM, 579
    vectors
        Property DSA, 580
distributivity, vector addition
    column vectors
        Property DVAC, 581
    vectors
        Property DVA, 582
DLDS (theorem), 583
DLTA (diagram), 584
DLTM (diagram), 585
DM (definition), 586
DM (notation), 587
DM (section), 588
DM (theorem), 589
DMAM (Property), 590
DMFE (theorem), 591
DMHP (subsection, section HP), 592
DMHP (theorem), 593
DMMP (theorem), 594
DMS3 (example), 595
DMST (theorem), 596
DNLT (theorem), 597
DNMMM (subsection, section PDM), 598
DP (theorem), 599
DRCM (theorem), 600
DRCMA (theorem), 601
DRCS (theorem), 602
DRMM (theorem), 603
DRO (example), 604
DRO (subsection, section PDM), 605
DROEM (subsection, section PDM), 606
DS (definition), 607
DS (notation), 608
DS (subsection, section PD), 609
DSA (Property), 610
DSAC (Property), 611
DSAM (Property), 612
DSD (theorem), 613
DSFB (theorem), 614
DSFOS (theorem), 615
DSLI (theorem), 616
DSM22 (example), 617
DSP4 (example), 618
DSZI (theorem), 619
DSZV (theorem), 620
DT (theorem), 621
DTSLS (diagram), 622
DUTM (example), 623
DVA (Property), 624
DVAC (Property), 625
DVM (theorem), 626
DVS (subsection, section D), 627
DZM (definition), 628
DZRC (theorem), 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 IS), 1044
EXC (subsection, section IVLT), 1045
EXC (subsection, section LC), 1046
EXC (subsection, section LDS), 1047
EXC (subsection, section LI), 1048
EXC (subsection, section LISS), 1049
EXC (subsection, section LT), 1050
EXC (subsection, section MINM), 1051
EXC (subsection, section MISLE), 1052
EXC (subsection, section MM), 1053
EXC (subsection, section MO), 1054
EXC (subsection, section MR), 1055
EXC (subsection, section NM), 1056
EXC (subsection, section O), 1057
EXC (subsection, section PD), 1058
EXC (subsection, section PDM), 1059
EXC (subsection, section PEE), 1060
EXC (subsection, section PSM), 1061
EXC (subsection, section RREF), 1062
EXC (subsection, section S), 1063
EXC (subsection, section SD), 1064
EXC (subsection, section SLT), 1065
EXC (subsection, section SS), 1066
EXC (subsection, section SSLE), 1067
EXC (subsection, section T), 1068
EXC (subsection, section TSS), 1069
EXC (subsection, section VO), 1070
EXC (subsection, section VR), 1071
EXC (subsection, section VS), 1072
EXC (subsection, section WILA), 1073
extended echelon form
    submatrices
        example SEEF, 1074
extended reduced row-echelon form
    properties
        theorem PEEF, 1075

F (archetype), 1076
F (definition), 1077
F (section), 1078
F (subsection, section F), 1079
FDV (example), 1080
FF (subsection, section F), 1081
FF8 (example), 1082
Fibonacci sequence
    example FSCF, 1083
field
    definition F, 1084
FIMP (theorem), 1085
finite field
    size 8
        example FF8, 1086
four subsets
    example FS1, 1087
    example FS2, 1088
four subspaces
    dimension
        theorem DFS, 1089
FRAN (example), 1090
free variables
    example CFV, 1091
free variables, number
    theorem FVCS, 1092
free, independent variables
    example FDV, 1093
FS (section), 1094
FS (subsection, section FS), 1095
FS (subsection, section SD), 1096
FS (theorem), 1097
FS1 (example), 1098
FS2 (example), 1099
FSAG (example), 1100
FSCF (example), 1101
FTMR (diagram), 1102
FTMR (theorem), 1103
FTMRA (diagram), 1104
FV (subsection, section TSS), 1105
FVCS (theorem), 1106

G (archetype), 1107
G (theorem), 1108
GE4 (example), 1109
GE6 (example), 1110
GEE (subsection, section IS), 1111
GEK (theorem), 1112
generalized eigenspace
    as kernel
        theorem GEK, 1113
    definition GES, 1114
    dimension
        theorem DGES, 1115
    dimension 4 domain
        example GE4, 1116
    dimension 6 domain
        example GE6, 1117
    invariant subspace
        theorem GESIS, 1118
    nilpotent restriction
        theorem RGEN, 1119
    nilpotent restrictions, dimension 6 domain
        example GENR6, 1120
    notation, 1121
generalized eigenspace decomposition
    theorem GESD, 1122
generalized eigenvector
    definition GEV, 1123
GENR6 (example), 1124
GES (definition), 1125
GES (notation), 1126
GESD (subsection, section JCF), 1127
GESD (theorem), 1128
GESIS (theorem), 1129
GEV (definition), 1130
GFDL (appendix), 1131
GLT (diagram), 1132
GME (definition), 1133
GME (notation), 1134
goldilocks
    theorem G, 1135
Gram-Schmidt
    column vectors
        theorem GSP, 1136
    three vectors
        example GSTV, 1137
gram-schmidt
    mathematica, 1138
GS (technique, section PT), 1139
GSP (subsection, section O), 1140
GSP (theorem), 1141
GSP.MMA (computation, section MMA), 1142
GSTV (example), 1143
GT (subsection, section PD), 1144

H (archetype), 1145
Hadamard Identity
    notation, 1146
Hadamard identity
    definition HID, 1147
Hadamard Inverse
    notation, 1148
Hadamard inverse
    definition HI, 1149
Hadamard Product
    Diagonalizable Matrices
        theorem DMHP, 1150
    notation, 1151
Hadamard product
    commutativity
        theorem HPC, 1152
    definition HP, 1153
    diagonal matrices
        theorem DMMP, 1154
    distributivity
        theorem HPDAA, 1155
    example HP, 1156
    identity
        theorem HPHID, 1157
    inverse
        theorem HPHI, 1158
    scalar matrix multiplication
        theorem HPSMM, 1159
hermitian
    definition HM, 1160
Hermitian matrix
    inner product
        theorem HMIP, 1161
HI (definition), 1162
HI (notation), 1163
HID (definition), 1164
HID (notation), 1165
HISAA (example), 1166
HISAD (example), 1167
HM (definition), 1168
HM (subsection, section MM), 1169
HMEM5 (example), 1170
HMIP (theorem), 1171
HMOE (theorem), 1172
HMRE (theorem), 1173
HMVEI (theorem), 1174
homogeneous system
    Archetype C
        example AHSAC, 1175
    consistent
        theorem HSC, 1176
    definition HS, 1177
    infinitely many solutions
        theorem HMVEI, 1178
homogeneous systems
    linear independence, 1179
HP (definition), 1180
HP (example), 1181
HP (notation), 1182
HP (section), 1183
HPC (theorem), 1184
HPDAA (theorem), 1185
HPDM (example), 1186
HPHI (theorem), 1187
HPHID (theorem), 1188
HPSMM (theorem), 1189
HS (definition), 1190
HSC (theorem), 1191
HSE (section), 1192
HUSAB (example), 1193

I (archetype), 1194
I (technique, section PT), 1195
IAP (example), 1196
IAR (example), 1197
IAS (example), 1198
IAV (example), 1199
ICBM (theorem), 1200
ICLT (theorem), 1201
identities
    technique PI, 1202
identity matrix
    determinant, 1203
    example IM, 1204
    notation, 1205
IDLT (definition), 1206
IDV (definition), 1207
IE (definition), 1208
IE (notation), 1209
IFDVS (theorem), 1210
IILT (theorem), 1211
ILT (definition), 1212
ILT (diagram), 1213
ILT (section), 1214
ILTB (theorem), 1215
ILTD (subsection, section ILT), 1216
ILTD (theorem), 1217
ILTIS (theorem), 1218
ILTLI (subsection, section ILT), 1219
ILTLI (theorem), 1220
ILTLT (theorem), 1221
ILTVR (example), 1222
IM (definition), 1223
IM (example), 1224
IM (notation), 1225
IM (subsection, section MISLE), 1226
IM11 (example), 1227
IMILT (theorem), 1228
IMP (definition), 1229
IMR (theorem), 1230
inconsistent linear systems
    theorem ISRN, 1231
independent, dependent variables
    definition IDV, 1232
indesxstring
    example SM2Z7, 1233
    example SSET, 1234
index
    eigenvalue
        definition IE, 1235
        notation, 1236
indexstring
    theorem DRCMA, 1237
    theorem OBUTR, 1238
    theorem UMCOB, 1239
induction
    technique I, 1240
infinite solution set
    example ISSI, 1241
infinite solutions, 3 × 4
    example IS, 1242
injective
    example IAP, 1243
    example IAR, 1244
    not
        example NIAO, 1245
        example NIAQ, 1246
        example NIAQR, 1247
    not, by dimension
        example NIDAU, 1248
    polynomials to matrices
        example IAV, 1249
injective linear transformation
    bases
        theorem ILTB, 1250
injective linear transformations
    dimension
        theorem ILTD, 1251
inner product
    anti-commutative
        theorem IPAC, 1252
    example CSIP, 1253
    norm
        theorem IPN, 1254
    notation, 1255
    positive
        theorem PIP, 1256
    scalar multiplication
        theorem IPSM, 1257
    vector addition
        theorem IPVA, 1258
integers
    mod p
        definition IMP, 1259
    mod p, field
        theorem FIMP, 1260
    mod 11
        example IM11, 1261
interpolating polynomial
    theorem IP, 1262
invariant subspace
    definition IS, 1263
    eigenspace, 1264
    eigenspaces
        example EIS, 1265
    example TIS, 1266
    Jordan block
        example ISJB, 1267
    kernels of powers
        theorem KPIS, 1268
inverse
    composition of linear transformations
        theorem ICLT, 1269
    example CMI, 1270
    example MI, 1271
    notation, 1272
    of a matrix, 1273
invertible linear transformation
    defined by invertible matrix
        theorem IMILT, 1274
invertible linear transformations
    composition
        theorem CIVLT, 1275
    computing
        example CIVLT, 1276
IP (definition), 1277
IP (notation), 1278
IP (subsection, section O), 1279
IP (theorem), 1280
IPAC (theorem), 1281
IPN (theorem), 1282
IPSM (theorem), 1283
IPVA (theorem), 1284
IS (definition), 1285
IS (example), 1286
IS (section), 1287
IS (subsection, section IS), 1288
ISJB (example), 1289
ISMR4 (example), 1290
ISMR6 (example), 1291
isomorphic
    multiple vector spaces
        example MIVS, 1292
    vector spaces
        example IVSAV, 1293
isomorphic vector spaces
    dimension
        theorem IVSED, 1294
    example TIVS, 1295
ISRN (theorem), 1296
ISSI (example), 1297
ITMT (theorem), 1298
IV (subsection, section IVLT), 1299
IVLT (definition), 1300
IVLT (section), 1301
IVLT (subsection, section IVLT), 1302
IVLT (subsection, section MR), 1303
IVS (definition), 1304
IVSAV (example), 1305
IVSED (theorem), 1306

J (archetype), 1307
JB (definition), 1308
JB (notation), 1309
JB4 (example), 1310
JCF (definition), 1311
JCF (section), 1312
JCF (subsection, section JCF), 1313
JCF10 (example), 1314
JCFLT (theorem), 1315
Jordan block
    definition JB, 1316
    nilpotent
        theorem NJB, 1317
    notation, 1318
    size 4
        example JB4, 1319
Jordan canonical form
    definition JCF, 1320
    size 10
        example JCF10, 1321

K (archetype), 1322
kernel
    injective linear transformation
        theorem KILT, 1323
    isomorphic to null space
        theorem KNSI, 1324
    linear transformation
        example NKAO, 1325
    notation, 1326
    of a linear transformation
        definition KLT, 1327
    pre-image, 1328
    subspace
        theorem KLTS, 1329
    trivial
        example TKAP, 1330
    via matrix representation
        example KVMR, 1331
KILT (theorem), 1332
KLT (definition), 1333
KLT (notation), 1334
KLT (subsection, section ILT), 1335
KLTS (theorem), 1336
KNSI (theorem), 1337
KPI (theorem), 1338
KPIS (theorem), 1339
KPLT (theorem), 1340
KPNLT (example), 1341
KPNLT (theorem), 1342
KVMR (example), 1343

L (archetype), 1344
L (technique, section PT), 1345
LA (subsection, section WILA), 1346
LC (definition), 1347
LC (section), 1348
LC (subsection, section LC), 1349
LC (technique, section PT), 1350
LCCV (definition), 1351
LCM (example), 1352
LDCAA (example), 1353
LDHS (example), 1354
LDP4 (example), 1355
LDRN (example), 1356
LDS (example), 1357
LDS (section), 1358
LDSS (subsection, section LDS), 1359
least squares
    minimizes residuals
        theorem LSMR, 1360
least squares solution
    definition LSS, 1361
left null space
    as row space, 1362
    definition LNS, 1363
    example LNS, 1364
    notation, 1365
    subspace
        theorem LNSMS, 1366
lemma
    technique LC, 1367
LI (definition), 1368
LI (section), 1369
LI (subsection, section LISS), 1370
LIC (example), 1371
LICAB (example), 1372
LICV (definition), 1373
LIHS (example), 1374
LIM32 (example), 1375
linear combination
    system of equations
        example ABLC, 1376
    definition LC, 1377
    definition LCCV, 1378
    example TLC, 1379
    linear transformation, 1380
    matrices
        example LCM, 1381
    system of equations
        example AALC, 1382
linear combinations
    solutions to linear systems
        theorem SLSLC, 1383
linear dependence
    more vectors than size
        theorem MVSLD, 1384
linear independence
    definition LI, 1385
    definition LICV, 1386
    homogeneous systems
        theorem LIVHS, 1387
    injective linear transformation
        theorem ILTLI, 1388
    matrices
        example LIM32, 1389
    orthogonal, 1390
    r and n
        theorem LIVRN, 1391
linear solve
    mathematica, 1392
    sage, 1393
linear system
    consistent
        theorem RCLS, 1394
    matrix representation
        definition MRLS, 1395
        notation, 1396
linear systems
    notation
        example MNSLE, 1397
        example NSLE, 1398
linear transformation
    polynomials to polynomials
        example LTPP, 1399
    addition
        definition LTA, 1400
        theorem MLTLT, 1401
        theorem SLTLT, 1402
    as matrix multiplication
        example ALTMM, 1403
    basis of range
        example BRLT, 1404
    checking
        example ALT, 1405
    composition
        definition LTC, 1406
        theorem CLTLT, 1407
    defined by a matrix
        example LTM, 1408
    defined on a basis
        example LTDB1, 1409
        example LTDB2, 1410
        example LTDB3, 1411
        theorem LTDB, 1412
    definition LT, 1413
    identity
        definition IDLT, 1414
    injection
        definition ILT, 1415
    inverse
        theorem ILTLT, 1416
    inverse of inverse
        theorem IILT, 1417
    invertible
        definition IVLT, 1418
        example AIVLT, 1419
    invertible, injective and surjective
        theorem ILTIS, 1420
    Jordan canonical form
        theorem JCFLT, 1421
    kernels of powers
        theorem KPLT, 1422
    linear combination
        theorem LTLC, 1423
    matrix of, 1424
        example MFLT, 1425
        example MOLT, 1426
    not
        example NLT, 1427
    not invertible
        example ANILT, 1428
    notation, 1429
    polynomials to matrices
        example LTPM, 1430
    rank plus nullity
        theorem RPNDD, 1431
    restriction
        definition LTR, 1432
        notation, 1433
    scalar multiple
        example SMLT, 1434
    scalar multiplication
        definition LTSM, 1435
    spanning range
        theorem SSRLT, 1436
    sum
        example STLT, 1437
    surjection
        definition SLT, 1438
    vector space of, 1439
    zero vector
        theorem LTTZZ, 1440
linear transformation inverse
    via matrix representation
        example ILTVR, 1441
linear transformation restriction
    on generalized eigenspace
        example LTRGE, 1442
linear transformations
    compositions
        example CTLT, 1443
    from matrices
        theorem MBLT, 1444
linearly dependent
    r < n
        example LDRN, 1445
    via homogeneous system
        example LDHS, 1446
linearly dependent columns
    Archetype A
        example LDCAA, 1447
linearly dependent set
    example LDS, 1448
    linear combinations within
        theorem DLDS, 1449
    polynomials
        example LDP4, 1450
linearly independent
    crazy vector space
        example LIC, 1451
    extending sets
        theorem ELIS, 1452
    polynomials
        example LIP4, 1453
    via homogeneous system
        example LIHS, 1454
linearly independent columns
    Archetype B
        example LICAB, 1455
linearly independent set
    example LIS, 1456
    example LLDS, 1457
LINM (subsection, section LI), 1458
LINSB (example), 1459
LIP4 (example), 1460
LIS (example), 1461
LISS (section), 1462
LISV (subsection, section LI), 1463
LIVHS (theorem), 1464
LIVRN (theorem), 1465
LLDS (example), 1466
LNS (definition), 1467
LNS (example), 1468
LNS (notation), 1469
LNS (subsection, section FS), 1470
LNSMS (theorem), 1471
lower triangular matrix
    definition LTM, 1472
LS.MMA (computation, section MMA), 1473
LS.SAGE (computation, section SAGE), 1474
LSMR (theorem), 1475
LSS (definition), 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 (archetype), 1500
M (chapter), 1501
M (definition), 1502
M (notation), 1503
MA (definition), 1504
MA (example), 1505
MA (notation), 1506
MACN (Property), 1507
MAF (Property), 1508
MAP (subsection, section SVD), 1509
mathematica
    gram-schmidt (computation), 1510
    linear solve (computation), 1511
    matrix entry (computation), 1512
    matrix inverse (computation), 1513
    matrix multiplication (computation), 1514
    null space (computation), 1515
    row reduce (computation), 1516
    transpose of a matrix (computation), 1517
    vector form of solutions (computation), 1518
    vector linear combinations (computation), 1519
mathematical language
    technique L, 1520
matrix
    addition
        definition MA, 1521
        notation, 1522
    augmented
        definition AM, 1523
    column space
        definition CSM, 1524
    complex conjugate
        example CCM, 1525
    definition M, 1526
    equality
        definition ME, 1527
        notation, 1528
    example AM, 1529
    identity
        definition IM, 1530
    inverse
        definition MI, 1531
    nonsingular
        definition NM, 1532
    notation, 1533
    of a linear transformation
        theorem MLTCV, 1534
    product
        example PTM, 1535
        example PTMEE, 1536
    product with vector
        definition MVP, 1537
    rectangular, 1538
    row space
        definition RSM, 1539
    scalar multiplication
        definition MSM, 1540
        notation, 1541
    singular, 1542
    square
        definition SQM, 1543
    submatrices
        example SS, 1544
    submatrix
        definition SM, 1545
    symmetric
        definition SYM, 1546
    transpose
        definition TM, 1547
    unitary
        definition UM, 1548
    unitary is invertible
        theorem UMI, 1549
    zero
        definition ZM, 1550
matrix addition
    example MA, 1551
matrix components
    notation, 1552
matrix entry
    mathematica, 1553
    sage, 1554
    ti83, 1555
    ti86, 1556
matrix inverse
    Archetype B, 1557
    computation
        theorem CINM, 1558
    mathematica, 1559
    nonsingular matrix
        theorem NI, 1560
    of a matrix inverse
        theorem MIMI, 1561
    one-sided
        theorem OSIS, 1562
    product
        theorem SS, 1563
    sage, 1564
    scalar multiple
        theorem MISM, 1565
    size 2 matrices
        theorem TTMI, 1566
    transpose
        theorem MIT, 1567
    uniqueness
        theorem MIU, 1568
matrix multiplication
    adjoints
        theorem MMAD, 1569
    associativity
        theorem MMA, 1570
    complex conjugation
        theorem MMCC, 1571
    definition MM, 1572
    distributivity
        theorem MMDAA, 1573
    entry-by-entry
        theorem EMP, 1574
    identity matrix
        theorem MMIM, 1575
    inner product
        theorem MMIP, 1576
    mathematica, 1577
    noncommutative
        example MMNC, 1578
    scalar matrix multiplication
        theorem MMSMM, 1579
    systems of linear equations
        theorem SLEMM, 1580
    transposes
        theorem MMT, 1581
    zero matrix
        theorem MMZM, 1582
matrix product
    as composition of linear transformations
        example MPMR, 1583
matrix representation
    basis of eigenvectors
        example MRBE, 1584
    composition of linear transformations
        theorem MRCLT, 1585
    definition MR, 1586
    invertible
        theorem IMR, 1587
    multiple of a linear transformation
        theorem MRMLT, 1588
    notation, 1589
    restriction to generalized eigenspace
        theorem MRRGE, 1590
    sum of linear transformations
        theorem MRSLT, 1591
    theorem FTMR, 1592
    upper triangular
        theorem UTMR, 1593
matrix representations
    converting with change-of-basis
        example MRCM, 1594
    example OLTTR, 1595
matrix scalar multiplication
    example MSM, 1596
matrix vector space
    dimension
        theorem DM, 1597
matrix-adjoint product
    eigenvalues, eigenvectors
        theorem EEMAP, 1598
matrix-vector product
    example MTV, 1599
    notation, 1600
MBC (example), 1601
MBLT (theorem), 1602
MC (notation), 1603
MCC (subsection, section MO), 1604
MCCN (Property), 1605
MCF (Property), 1606
MCN (definition), 1607
MCN (subsection, section CNO), 1608
MCSM (example), 1609
MCT (theorem), 1610
MD (chapter), 1611
ME (definition), 1612
ME (notation), 1613
ME (subsection, section PEE), 1614
ME (technique, section PT), 1615
ME (theorem), 1616
ME.MMA (computation, section MMA), 1617
ME.SAGE (computation, section SAGE), 1618
ME.TI83 (computation, section TI83), 1619
ME.TI86 (computation, section TI86), 1620
MEASM (subsection, section MO), 1621
MFLT (example), 1622
MI (definition), 1623
MI (example), 1624
MI (notation), 1625
MI.MMA (computation, section MMA), 1626
MI.SAGE (computation, section SAGE), 1627
MICN (Property), 1628
MIF (Property), 1629
MIMI (theorem), 1630
MINM (section), 1631
MISLE (section), 1632
MISM (theorem), 1633
MIT (theorem), 1634
MIU (theorem), 1635
MIVS (example), 1636
MLT (subsection, section LT), 1637
MLTCV (theorem), 1638
MLTLT (theorem), 1639
MM (definition), 1640
MM (section), 1641
MM (subsection, section MM), 1642
MM.MMA (computation, section MMA), 1643
MMA (section), 1644
MMA (theorem), 1645
MMAD (theorem), 1646
MMCC (theorem), 1647
MMDAA (theorem), 1648
MMEE (subsection, section MM), 1649
MMIM (theorem), 1650
MMIP (theorem), 1651
MMNC (example), 1652
MMSMM (theorem), 1653
MMT (theorem), 1654
MMZM (theorem), 1655
MNEM (theorem), 1656
MNSLE (example), 1657
MO (section), 1658
MOLT (example), 1659
more variables than equations
    example OSGMD, 1660
    theorem CMVEI, 1661
MPMR (example), 1662
MR (definition), 1663
MR (notation), 1664
MR (section), 1665
MRBE (example), 1666
MRCB (theorem), 1667
MRCLT (diagram), 1668
MRCLT (theorem), 1669
MRCM (example), 1670
MRLS (definition), 1671
MRLS (notation), 1672
MRMLT (theorem), 1673
MRRGE (theorem), 1674
MRS (subsection, section CB), 1675
MRSLT (theorem), 1676
MSCN (example), 1677
MSM (definition), 1678
MSM (example), 1679
MSM (notation), 1680
MTV (example), 1681
multiplicative associativity
    complex numbers
        Property MACN, 1682
multiplicative closure
    complex numbers
        Property MCCN, 1683
    field
        Property MCF, 1684
multiplicative commutativity
    complex numbers
        Property CMCN, 1685
multiplicative inverse
    complex numbers
        Property MICN, 1686
MVNSE (subsection, section RREF), 1687
MVP (definition), 1688
MVP (notation), 1689
MVP (subsection, section MM), 1690
MVSLD (theorem), 1691
MWIAA (example), 1692

N (archetype), 1693
N (subsection, section O), 1694
N (technique, section PT), 1695
NDMS4 (example), 1696
negation of statements
    technique N, 1697
NEM (theorem), 1698
NI (theorem), 1699
NIAO (example), 1700
NIAQ (example), 1701
NIAQR (example), 1702
NIDAU (example), 1703
nilpotent
    linear transformation
        definition NLT, 1704
NILT (diagram), 1705
NJB (theorem), 1706
NJB5 (example), 1707
NKAO (example), 1708
NLT (definition), 1709
NLT (example), 1710
NLT (section), 1711
NLT (subsection, section NLT), 1712
NLTFO (subsection, section LT), 1713
NM (definition), 1714
NM (example), 1715
NM (section), 1716
NM (subsection, section NM), 1717
NM (subsection, section OD), 1718
NM62 (example), 1719
NM64 (example), 1720
NM83 (example), 1721
NME1 (theorem), 1722
NME2 (theorem), 1723
NME3 (theorem), 1724
NME4 (theorem), 1725
NME5 (theorem), 1726
NME6 (theorem), 1727
NME7 (theorem), 1728
NME8 (theorem), 1729
NME9 (theorem), 1730
NMI (subsection, section MINM), 1731
NMLIC (theorem), 1732
NMPEM (theorem), 1733
NMRRI (theorem), 1734
NMTNS (theorem), 1735
NMUS (theorem), 1736
NOILT (theorem), 1737
NOLT (definition), 1738
NOLT (notation), 1739
NOM (definition), 1740
NOM (notation), 1741
nonsingular
    columns as basis
        theorem CNMB, 1742
nonsingular matrices
    linearly independent columns
        theorem NMLIC, 1743
nonsingular matrix
    Archetype B
        example NM, 1744
    column space, 1745
    elementary matrices
        theorem NMPEM, 1746
    equivalences
        theorem NME1, 1747
        theorem NME2, 1748
        theorem NME3, 1749
        theorem NME4, 1750
        theorem NME5, 1751
        theorem NME6, 1752
        theorem NME7, 1753
        theorem NME8, 1754
        theorem NME9, 1755
    matrix inverse, 1756
    null space
        example NSNM, 1757
    nullity, 1758
    product of nonsingular matrices
        theorem NPNT, 1759
    rank
        theorem RNNM, 1760
    row-reduced
        theorem NMRRI, 1761
    trivial null space
        theorem NMTNS, 1762
    unique solutions
        theorem NMUS, 1763
nonsingular matrix, row-reduced
    example NSR, 1764
norm
    example CNSV, 1765
    inner product, 1766
    notation, 1767
normal matrix
    definition NRML, 1768
    example ANM, 1769
    orthonormal basis, 1770
notation
    A, 1771
    AM, 1772
    AME, 1773
    C, 1774
    CCCV, 1775
    CCM, 1776
    CCN, 1777
    CNA, 1778
    CNE, 1779
    CNM, 1780
    CSM, 1781
    CV, 1782
    CVA, 1783
    CVC, 1784
    CVE, 1785
    CVSM, 1786
    D, 1787
    DM, 1788
    DS, 1789
    ELEM, 1790
    ES, 1791
    GES, 1792
    GME, 1793
    HI, 1794
    HID, 1795
    HP, 1796
    IE, 1797
    IM, 1798
    IP, 1799
    JB, 1800
    KLT, 1801
    LNS, 1802
    LT, 1803
    LTR, 1804
    M, 1805
    MA, 1806
    MC, 1807
    ME, 1808
    MI, 1809
    MR, 1810
    MRLS, 1811
    MSM, 1812
    MVP, 1813
    NOLT, 1814
    NOM, 1815
    NSM, 1816
    NV, 1817
    RLT, 1818
    RO, 1819
    ROLT, 1820
    ROM, 1821
    RREFA, 1822
    RSM, 1823
    SC, 1824
    SE, 1825
    SETM, 1826
    SI, 1827
    SM, 1828
    SRM, 1829
    SSET, 1830
    SSV, 1831
    SU, 1832
    SUV, 1833
    T, 1834
    TM, 1835
    VR, 1836
    VSCV, 1837
    VSM, 1838
    ZCV, 1839
    ZM, 1840
notation for a linear system
    example NSE, 1841
NPNT (theorem), 1842
NRFO (subsection, section MR), 1843
NRML (definition), 1844
NRREF (example), 1845
NS.MMA (computation, section MMA), 1846
NSAO (example), 1847
NSAQ (example), 1848
NSAQR (example), 1849
NSC2A (example), 1850
NSC2S (example), 1851
NSC2Z (example), 1852
NSDAT (example), 1853
NSDS (example), 1854
NSE (example), 1855
NSEAI (example), 1856
NSLE (example), 1857
NSLIL (example), 1858
NSM (definition), 1859
NSM (notation), 1860
NSM (subsection, section HSE), 1861
NSMS (theorem), 1862
NSNM (example), 1863
NSNM (subsection, section NM), 1864
NSR (example), 1865
NSS (example), 1866
NSSLI (subsection, section LI), 1867
Null space
    as a span
        example NSDS, 1868
null space
    Archetype I
        example NSEAI, 1869
    basis
        theorem BNS, 1870
    computation
        example CNS1, 1871
        example CNS2, 1872
    isomorphic to kernel, 1873
    linearly independent basis
        example LINSB, 1874
    mathematica, 1875
    matrix
        definition NSM, 1876
    nonsingular matrix, 1877
    notation, 1878
    singular matrix, 1879
    spanning set
        example SSNS, 1880
        theorem SSNS, 1881
    subspace
        theorem NSMS, 1882
null space span, linearly independent
    Archetype L
        example NSLIL, 1883
nullity
    computing, 1884
    injective linear transformation
        theorem NOILT, 1885
    linear transformation
        definition NOLT, 1886
    matrix, 1887
        definition NOM, 1888
    notation, 1889, 1890
    square matrix, 1891
NV (definition), 1892
NV (notation), 1893
NVM (theorem), 1894

O (archetype), 1895
O (Property), 1896
O (section), 1897
OBC (subsection, section B), 1898
OBNM (theorem), 1899
OBUTR (theorem), 1900
OC (Property), 1901
OCN (Property), 1902
OD (section), 1903
OD (subsection, section OD), 1904
OD (theorem), 1905
OF (Property), 1906
OLTTR (example), 1907
OM (Property), 1908
one
    column vectors
        Property OC, 1909
    complex numbers
        Property OCN, 1910
    field
        Property OF, 1911
    matrices
        Property OM, 1912
    vectors
        Property O, 1913
ONFV (example), 1914
ONS (definition), 1915
ONTV (example), 1916
orthogonal
    linear independence
        theorem OSLI, 1917
    set
        example AOS, 1918
    set of vectors
        definition OSV, 1919
    vector pairs
        definition OV, 1920
orthogonal vectors
    example TOV, 1921
orthonormal
    definition ONS, 1922
    matrix columns
        example OSMC, 1923
orthonormal basis
    normal matrix
        theorem OBNM, 1924
orthonormal diagonalization
    theorem OD, 1925
orthonormal set
    four vectors
        example ONFV, 1926
    three vectors
        example ONTV, 1927
OSGMD (example), 1928
OSIS (theorem), 1929
OSLI (theorem), 1930
OSMC (example), 1931
OSV (definition), 1932
OV (definition), 1933
OV (subsection, section O), 1934

P (appendix), 1935
P (archetype), 1936
P (technique, section PT), 1937
particular solutions
    example PSHS, 1938
PCNA (theorem), 1939
PCVS (example), 1940
PD (section), 1941
PDM (section), 1942
PDM (theorem), 1943
PEE (section), 1944
PEEF (theorem), 1945
PI (definition), 1946
PI (subsection, section LT), 1947
PI (technique, section PT), 1948
PIP (theorem), 1949
PM (example), 1950
PM (subsection, section EE), 1951
PMI (subsection, section MISLE), 1952
PMM (subsection, section MM), 1953
PMR (subsection, section MR), 1954
PNLT (subsection, section NLT), 1955
POD (section), 1956
polar decomposition
    theorem PDM, 1957
polynomial
    of a matrix
        example PM, 1958
polynomial vector space
    dimension
        theorem DP, 1959
positive semi-definite
    creating
        theorem CPSM, 1960
positive semi-definite matrix
    definition PSM, 1961
    eigenvalues
        theorem EPSM, 1962
practice
    technique P, 1963
pre-image
    definition PI, 1964
    kernel
        theorem KPI, 1965
pre-images
    example SPIAS, 1966
principal axis theorem, 1967
product of triangular matrices
    theorem PTMT, 1968
Property
    AA, 1969
    AAC, 1970
    AACN, 1971
    AAF, 1972
    AAM, 1973
    AC, 1974
    ACC, 1975
    ACCN, 1976
    ACF, 1977
    ACM, 1978
    AI, 1979
    AIC, 1980
    AICN, 1981
    AIF, 1982
    AIM, 1983
    C, 1984
    CACN, 1985
    CAF, 1986
    CC, 1987
    CM, 1988
    CMCN, 1989
    CMF, 1990
    DCN, 1991
    DF, 1992
    DMAM, 1993
    DSA, 1994
    DSAC, 1995
    DSAM, 1996
    DVA, 1997
    DVAC, 1998
    MACN, 1999
    MAF, 2000
    MCCN, 2001
    MCF, 2002
    MICN, 2003
    MIF, 2004
    O, 2005
    OC, 2006
    OCN, 2007
    OF, 2008
    OM, 2009
    SC, 2010
    SCC, 2011
    SCM, 2012
    SMA, 2013
    SMAC, 2014
    SMAM, 2015
    Z, 2016
    ZC, 2017
    ZCN, 2018
    ZF, 2019
    ZM, 2020
PSHS (example), 2021
PSHS (subsection, section LC), 2022
PSM (definition), 2023
PSM (section), 2024
PSM (subsection, section PSM), 2025
PSM (subsection, section SD), 2026
PSMSR (theorem), 2027
PSPHS (theorem), 2028
PSS (subsection, section SSLE), 2029
PSSD (theorem), 2030
PSSLS (theorem), 2031
PT (section), 2032
PTFP (example), 2033
PTM (example), 2034
PTMEE (example), 2035
PTMT (theorem), 2036

Q (archetype), 2037

R (archetype), 2038
R (chapter), 2039
R.SAGE (computation, section SAGE), 2040
range
    full
        example FRAN, 2041
    isomorphic to column space
        theorem RCSI, 2042
    linear transformation
        example RAO, 2043
    notation, 2044
    of a linear transformation
        definition RLT, 2045
    pre-image
        theorem RPI, 2046
    subspace
        theorem RLTS, 2047
    surjective linear transformation
        theorem RSLT, 2048
    via matrix representation
        example RVMR, 2049
rank
    computing
        theorem CRN, 2050
    linear transformation
        definition ROLT, 2051
    matrix
        definition ROM, 2052
        example RNM, 2053
    notation, 2054, 2055
    of transpose
        example RRTI, 2056
    square matrix
        example RNSM, 2057
    surjective linear transformation
        theorem ROSLT, 2058
    transpose
        theorem RMRT, 2059
rank one decomposition
    size 2
        example ROD2, 2060
    size 4
        example ROD4, 2061
    theorem ROD, 2062
rank+nullity
    theorem RPNC, 2063
RAO (example), 2064
RCLS (theorem), 2065
RCSI (theorem), 2066
RD (subsection, section VS), 2067
RDS (theorem), 2068
READ (subsection, section B), 2069
READ (subsection, section CB), 2070
READ (subsection, section CRS), 2071
READ (subsection, section D), 2072
READ (subsection, section DM), 2073
READ (subsection, section EE), 2074
READ (subsection, section FS), 2075
READ (subsection, section HSE), 2076
READ (subsection, section ILT), 2077
READ (subsection, section IVLT), 2078
READ (subsection, section LC), 2079
READ (subsection, section LDS), 2080
READ (subsection, section LI), 2081
READ (subsection, section LISS), 2082
READ (subsection, section LT), 2083
READ (subsection, section MINM), 2084
READ (subsection, section MISLE), 2085
READ (subsection, section MM), 2086
READ (subsection, section MO), 2087
READ (subsection, section MR), 2088
READ (subsection, section NM), 2089
READ (subsection, section O), 2090
READ (subsection, section PD), 2091
READ (subsection, section PDM), 2092
READ (subsection, section PEE), 2093
READ (subsection, section RREF), 2094
READ (subsection, section S), 2095
READ (subsection, section SD), 2096
READ (subsection, section SLT), 2097
READ (subsection, section SS), 2098
READ (subsection, section SSLE), 2099
READ (subsection, section TSS), 2100
READ (subsection, section VO), 2101
READ (subsection, section VR), 2102
READ (subsection, section VS), 2103
READ (subsection, section WILA), 2104
reduced row-echelon form
    analysis
        notation, 2105
    definition RREF, 2106
    example NRREF, 2107
    example RREF, 2108
    extended
        definition EEF, 2109
    notation
        example RREFN, 2110
    unique
        theorem RREFU, 2111
reducing a span
    example RSC5, 2112
relation of linear dependence
    definition RLD, 2113
    definition RLDCV, 2114
REM (definition), 2115
REMEF (theorem), 2116
REMES (theorem), 2117
REMRS (theorem), 2118
RES (example), 2119
RGEN (theorem), 2120
rings
    sage, 2121
RLD (definition), 2122
RLDCV (definition), 2123
RLT (definition), 2124
RLT (notation), 2125
RLT (subsection, section IS), 2126
RLT (subsection, section SLT), 2127
RLTS (theorem), 2128
RMRT (theorem), 2129
RNLT (subsection, section IVLT), 2130
RNM (example), 2131
RNM (subsection, section D), 2132
RNNM (subsection, section D), 2133
RNNM (theorem), 2134
RNSM (example), 2135
RO (definition), 2136
RO (notation), 2137
RO (subsection, section RREF), 2138
ROD (section), 2139
ROD (theorem), 2140
ROD2 (example), 2141
ROD4 (example), 2142
ROLT (definition), 2143
ROLT (notation), 2144
ROM (definition), 2145
ROM (notation), 2146
ROSLT (theorem), 2147
row operations
    definition RO, 2148
    elementary matrices, 2149, 2150
    notation, 2151
row reduce
    mathematica, 2152
    sage, 2153
    ti83, 2154
    ti86, 2155
row space
    Archetype I
        example RSAI, 2156
    as column space, 2157
    basis
        example RSB, 2158
        theorem BRS, 2159
    matrix, 2160
    notation, 2161
    row-equivalent matrices
        theorem REMRS, 2162
    subspace
        theorem RSMS, 2163
row-equivalent matrices
    definition REM, 2164
    example TREM, 2165
    row space, 2166
    row spaces
        example RSREM, 2167
    theorem REMES, 2168
row-reduce
    the verb
        definition RR, 2169
row-reduced matrices
    theorem REMEF, 2170
RPI (theorem), 2171
RPNC (theorem), 2172
RPNDD (theorem), 2173
RR (definition), 2174
RR.MMA (computation, section MMA), 2175
RR.SAGE (computation, section SAGE), 2176
RR.TI83 (computation, section TI83), 2177
RR.TI86 (computation, section TI86), 2178
RREF (definition), 2179
RREF (example), 2180
RREF (section), 2181
RREF (subsection, section RREF), 2182
RREFA (notation), 2183
RREFN (example), 2184
RREFU (theorem), 2185
RRTI (example), 2186
RS (example), 2187
RSAI (example), 2188
RSB (example), 2189
RSC4 (example), 2190
RSC5 (example), 2191
RSLT (theorem), 2192
RSM (definition), 2193
RSM (notation), 2194
RSM (subsection, section CRS), 2195
RSMS (theorem), 2196
RSNS (example), 2197
RSREM (example), 2198
RT (subsection, section PD), 2199
RVMR (example), 2200

S (archetype), 2201
S (definition), 2202
S (example), 2203
S (section), 2204
SAA (example), 2205
SAB (example), 2206
SABMI (example), 2207
SAE (example), 2208
sage
    eigenspaces (computation), 2209
    linear solve (computation), 2210
    matrix entry (computation), 2211
    matrix inverse (computation), 2212
    rings (computation), 2213
    row reduce (computation), 2214
    transpose of a matrix (computation), 2215
    vector linear combinations (computation), 2216
SAGE (section), 2217
SAN (example), 2218
SAR (example), 2219
SAS (section), 2220
SAV (example), 2221
SC (definition), 2222
SC (example), 2223
SC (notation), 2224
SC (Property), 2225
SC (subsection, section S), 2226
SC (subsection, section SET), 2227
SC3 (example), 2228
SCAA (example), 2229
SCAB (example), 2230
SCAD (example), 2231
scalar closure
    column vectors
        Property SCC, 2232
    matrices
        Property SCM, 2233
    vectors
        Property SC, 2234
scalar multiple
    matrix inverse, 2235
scalar multiplication
    zero scalar
        theorem ZSSM, 2236
    zero vector
        theorem ZVSM, 2237
    zero vector result
        theorem SMEZV, 2238
scalar multiplication associativity
    column vectors
        Property SMAC, 2239
    matrices
        Property SMAM, 2240
    vectors
        Property SMA, 2241
SCB (theorem), 2242
SCC (Property), 2243
SCM (Property), 2244
SD (section), 2245
SDS (example), 2246
SE (definition), 2247
SE (notation), 2248
secret sharing
    6 ways
        example SS6W, 2249
SEE (example), 2250
SEEF (example), 2251
SER (theorem), 2252
set
    cardinality
        definition C, 2253
        example CS, 2254
        notation, 2255
    complement
        definition SC, 2256
        example SC, 2257
        notation, 2258
    definition SET, 2259
    empty
        definition ES, 2260
    equality
        definition SE, 2261
        notation, 2262
    intersection
        definition SI, 2263
        example SI, 2264
        notation, 2265
    membership
        example SETM, 2266
        notation, 2267
    size, 2268
    subset, 2269
    union
        definition SU, 2270
        example SU, 2271
        notation, 2272
SET (definition), 2273
SET (section), 2274
SETM (example), 2275
SETM (notation), 2276
shoes, 2277
SHS (subsection, section HSE), 2278
SI (definition), 2279
SI (example), 2280
SI (notation), 2281
SI (subsection, section IVLT), 2282
SIM (definition), 2283
similar matrices
    equal eigenvalues
        example EENS, 2284
    eual eigenvalues
        theorem SMEE, 2285
    example SMS3, 2286
    example SMS5, 2287
similarity
    definition SIM, 2288
    equivalence relation
        theorem SER, 2289
singular matrix
    Archetype A
        example S, 2290
    null space
        example NSS, 2291
singular matrix, row-reduced
    example SRR, 2292
singular value decomposition
    theorem SVD, 2293
singular values
    definition SV, 2294
SLE (chapter), 2295
SLE (definition), 2296
SLE (subsection, section SSLE), 2297
SLELT (subsection, section IVLT), 2298
SLEMM (theorem), 2299
SLSLC (theorem), 2300
SLT (definition), 2301
SLT (section), 2302
SLTB (theorem), 2303
SLTD (subsection, section SLT), 2304
SLTD (theorem), 2305
SLTLT (theorem), 2306
SM (definition), 2307
SM (notation), 2308
SM (subsection, section SD), 2309
SM2Z7 (example), 2310
SM32 (example), 2311
SMA (Property), 2312
SMAC (Property), 2313
SMAM (Property), 2314
SMEE (theorem), 2315
SMEZV (theorem), 2316
SMLT (example), 2317
SMS (theorem), 2318
SMS3 (example), 2319
SMS5 (example), 2320
SMZD (theorem), 2321
SMZE (theorem), 2322
SNCM (theorem), 2323
SO (subsection, section SET), 2324
socks, 2325
SOL (subsection, section B), 2326
SOL (subsection, section CB), 2327
SOL (subsection, section CRS), 2328
SOL (subsection, section D), 2329
SOL (subsection, section DM), 2330
SOL (subsection, section EE), 2331
SOL (subsection, section F), 2332
SOL (subsection, section FS), 2333
SOL (subsection, section HSE), 2334
SOL (subsection, section ILT), 2335
SOL (subsection, section IVLT), 2336
SOL (subsection, section LC), 2337
SOL (subsection, section LDS), 2338
SOL (subsection, section LI), 2339
SOL (subsection, section LISS), 2340
SOL (subsection, section LT), 2341
SOL (subsection, section MINM), 2342
SOL (subsection, section MISLE), 2343
SOL (subsection, section MM), 2344
SOL (subsection, section MO), 2345
SOL (subsection, section MR), 2346
SOL (subsection, section NM), 2347
SOL (subsection, section O), 2348
SOL (subsection, section PD), 2349
SOL (subsection, section PDM), 2350
SOL (subsection, section PEE), 2351
SOL (subsection, section RREF), 2352
SOL (subsection, section S), 2353
SOL (subsection, section SD), 2354
SOL (subsection, section SLT), 2355
SOL (subsection, section SS), 2356
SOL (subsection, section SSLE), 2357
SOL (subsection, section T), 2358
SOL (subsection, section TSS), 2359
SOL (subsection, section VO), 2360
SOL (subsection, section VR), 2361
SOL (subsection, section VS), 2362
SOL (subsection, section WILA), 2363
solution set
    Archetype A
        example SAA, 2364
    archetype E
        example SAE, 2365
    theorem PSPHS, 2366
solution set of a linear system
    definition SSSLE, 2367
solution sets
    possibilities
        theorem PSSLS, 2368
solution to a linear system
    definition SSLE, 2369
solution vector
    definition SOLV, 2370
SOLV (definition), 2371
solving homogeneous system
    Archetype A
        example HISAA, 2372
    Archetype B
        example HUSAB, 2373
    Archetype D
        example HISAD, 2374
solving nonlinear equations
    example STNE, 2375
SP4 (example), 2376
span
    basic
        example ABS, 2377
    basis
        theorem BS, 2378
    definition SS, 2379
    definition SSCV, 2380
    improved
        example IAS, 2381
    notation, 2382
    reducing
        example RSC4, 2383
    reduction
        example RS, 2384
    removing vectors
        example COV, 2385
    reworking elements
        example RES, 2386
    set of polynomials
        example SSP, 2387
    subspace
        theorem SSS, 2388
span of columns
    Archetype A
        example SCAA, 2389
    Archetype B
        example SCAB, 2390
    Archetype D
        example SCAD, 2391
spanning set
    crazy vector space
        example SSC, 2392
    definition TSVS, 2393
    matrices
        example SSM22, 2394
    more vectors
        theorem SSLD, 2395
    polynomials
        example SSP4, 2396
SPIAS (example), 2397
SQM (definition), 2398
square root
    eigenvalues, eigenspaces
        theorem EESR, 2399
    matrix
        definition SRM, 2400
        notation, 2401
    positive semi-definite matrix
        theorem PSMSR, 2402
    unique
        theorem USR, 2403
SR (section), 2404
SRM (definition), 2405
SRM (notation), 2406
SRM (subsection, section SR), 2407
SRR (example), 2408
SS (definition), 2409
SS (example), 2410
SS (section), 2411
SS (subsection, section LISS), 2412
SS (theorem), 2413
SS6W (example), 2414
SSC (example), 2415
SSCV (definition), 2416
SSET (definition), 2417
SSET (example), 2418
SSET (notation), 2419
SSLD (theorem), 2420
SSLE (definition), 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
SSSLE (definition), 2431
SSSLT (subsection, section SLT), 2432
SSV (notation), 2433
SSV (subsection, section SS), 2434
standard unit vector
    notation, 2435
starting proofs
    technique GS, 2436
STLT (example), 2437
STNE (example), 2438
SU (definition), 2439
SU (example), 2440
SU (notation), 2441
submatrix
    notation, 2442
subset
    definition SSET, 2443
    notation, 2444
subspace
    as null space
        example RSNS, 2445
    characterized
        example ASC, 2446
    definition S, 2447
    in {P}_{4}
        example SP4, 2448
    not, additive closure
        example NSC2A, 2449
    not, scalar closure
        example NSC2S, 2450
    not, zero vector
        example NSC2Z, 2451
    testing
        theorem TSS, 2452
    trivial
        definition TS, 2453
    verification
        example SC3, 2454
        example SM32, 2455
subspaces
    equal dimension
        theorem EDYES, 2456
surjective
    Archetype N
        example SAN, 2457
    example SAR, 2458
    not
        example NSAQ, 2459
        example NSAQR, 2460
    not, Archetype O
        example NSAO, 2461
    not, by dimension
        example NSDAT, 2462
    polynomials to matrices
        example SAV, 2463
surjective linear transformation
    bases
        theorem SLTB, 2464
surjective linear transformations
    dimension
        theorem SLTD, 2465
SUV (definition), 2466
SUV (notation), 2467
SUVB (theorem), 2468
SUVOS (example), 2469
SV (definition), 2470
SVD (section), 2471
SVD (subsection, section SVD), 2472
SVD (theorem), 2473
SVP4 (example), 2474
SYM (definition), 2475
SYM (example), 2476
symmetric matrices
    theorem SMS, 2477
symmetric matrix
    example SYM, 2478
system of equations
    vector equality
        example VESE, 2479
system of linear equations
    definition SLE, 2480

T (archetype), 2481
T (definition), 2482
T (notation), 2483
T (part), 2484
T (section), 2485
T (technique, section PT), 2486
TCSD (example), 2487
TD (section), 2488
TD (subsection, section TD), 2489
TD (theorem), 2490
TD4 (example), 2491
TDEE (theorem), 2492
TDEE6 (example), 2493
TDSSE (example), 2494
TDSSE (subsection, section TD), 2495
technique
    C, 2496
    CD, 2497
    CP, 2498
    CV, 2499
    D, 2500
    DC, 2501
    E, 2502
    GS, 2503
    I, 2504
    L, 2505
    LC, 2506
    ME, 2507
    N, 2508
    P, 2509
    PI, 2510
    T, 2511
    U, 2512
theorem
    AA, 2513
    AIP, 2514
    AISM, 2515
    AIU, 2516
    AMA, 2517
    AMSM, 2518
    BCS, 2519
    BIS, 2520
    BNS, 2521
    BRS, 2522
    BS, 2523
    CB, 2524
    CCM, 2525
    CCRA, 2526
    CCRM, 2527
    CCT, 2528
    CFDVS, 2529
    CFNLT, 2530
    CHT, 2531
    CILTI, 2532
    CINM, 2533
    CIVLT, 2534
    CLI, 2535
    CLTLT, 2536
    CMVEI, 2537
    CNMB, 2538
    COB, 2539
    CPSM, 2540
    CRMA, 2541
    CRMSM, 2542
    CRN, 2543
    CRSM, 2544
    CRVA, 2545
    CSCS, 2546
    CSLTS, 2547
    CSMS, 2548
    CSNM, 2549
    CSRN, 2550
    CSRST, 2551
    CSS, 2552
    CUMOS, 2553
    DC, 2554
    DCM, 2555
    DCP, 2556
    DEC, 2557
    DED, 2558
    DEM, 2559
    DEMMM, 2560
    DER, 2561
    DERC, 2562
    DFS, 2563
    DGES, 2564
    DIM, 2565
    DLDS, 2566
    DM, 2567
    DMFE, 2568
    DMHP, 2569
    DMMP, 2570
    DMST, 2571
    DNLT, 2572
    DP, 2573
    DRCM, 2574
    DRCMA, 2575
    DRCS, 2576
    DRMM, 2577
    DSD, 2578
    DSFB, 2579
    DSFOS, 2580
    DSLI, 2581
    DSZI, 2582
    DSZV, 2583
    DT, 2584
    DVM, 2585
    DZRC, 2586
    EDELI, 2587
    EDYES, 2588
    EEMAP, 2589
    EER, 2590
    EESR, 2591
    EIM, 2592
    EIS, 2593
    ELIS, 2594
    EMDRO, 2595
    EMHE, 2596
    EMMVP, 2597
    EMN, 2598
    EMNS, 2599
    EMP, 2600
    EMRCP, 2601
    EMS, 2602
    ENLT, 2603
    EOMP, 2604
    EOPSS, 2605
    EPM, 2606
    EPSM, 2607
    ERMCP, 2608
    ESMM, 2609
    ETM, 2610
    FIMP, 2611
    FS, 2612
    FTMR, 2613
    FVCS, 2614
    G, 2615
    GEK, 2616
    GESD, 2617
    GESIS, 2618
    GSP, 2619
    HMIP, 2620
    HMOE, 2621
    HMRE, 2622
    HMVEI, 2623
    HPC, 2624
    HPDAA, 2625
    HPHI, 2626
    HPHID, 2627
    HPSMM, 2628
    HSC, 2629
    ICBM, 2630
    ICLT, 2631
    IFDVS, 2632
    IILT, 2633
    ILTB, 2634
    ILTD, 2635
    ILTIS, 2636
    ILTLI, 2637
    ILTLT, 2638
    IMILT, 2639
    IMR, 2640
    IP, 2641
    IPAC, 2642
    IPN, 2643
    IPSM, 2644
    IPVA, 2645
    ISRN, 2646
    ITMT, 2647
    IVSED, 2648
    JCFLT, 2649
    KILT, 2650
    KLTS, 2651
    KNSI, 2652
    KPI, 2653
    KPIS, 2654
    KPLT, 2655
    KPNLT, 2656
    LIVHS, 2657
    LIVRN, 2658
    LNSMS, 2659
    LSMR, 2660
    LTDB, 2661
    LTLC, 2662
    LTTZZ, 2663
    MBLT, 2664
    MCT, 2665
    ME, 2666
    MIMI, 2667
    MISM, 2668
    MIT, 2669
    MIU, 2670
    MLTCV, 2671
    MLTLT, 2672
    MMA, 2673
    MMAD, 2674
    MMCC, 2675
    MMDAA, 2676
    MMIM, 2677
    MMIP, 2678
    MMSMM, 2679
    MMT, 2680
    MMZM, 2681
    MNEM, 2682
    MRCB, 2683
    MRCLT, 2684
    MRMLT, 2685
    MRRGE, 2686
    MRSLT, 2687
    MVSLD, 2688
    NEM, 2689
    NI, 2690
    NJB, 2691
    NME1, 2692
    NME2, 2693
    NME3, 2694
    NME4, 2695
    NME5, 2696
    NME6, 2697
    NME7, 2698
    NME8, 2699
    NME9, 2700
    NMLIC, 2701
    NMPEM, 2702
    NMRRI, 2703
    NMTNS, 2704
    NMUS, 2705
    NOILT, 2706
    NPNT, 2707
    NSMS, 2708
    NVM, 2709
    OBNM, 2710
    OBUTR, 2711
    OD, 2712
    OSIS, 2713
    OSLI, 2714
    PCNA, 2715
    PDM, 2716
    PEEF, 2717
    PIP, 2718
    PSMSR, 2719
    PSPHS, 2720
    PSSD, 2721
    PSSLS, 2722
    PTMT, 2723
    RCLS, 2724
    RCSI, 2725
    RDS, 2726
    REMEF, 2727
    REMES, 2728
    REMRS, 2729
    RGEN, 2730
    RLTS, 2731
    RMRT, 2732
    RNNM, 2733
    ROD, 2734
    ROSLT, 2735
    RPI, 2736
    RPNC, 2737
    RPNDD, 2738
    RREFU, 2739
    RSLT, 2740
    RSMS, 2741
    SCB, 2742
    SER, 2743
    SLEMM, 2744
    SLSLC, 2745
    SLTB, 2746
    SLTD, 2747
    SLTLT, 2748
    SMEE, 2749
    SMEZV, 2750
    SMS, 2751
    SMZD, 2752
    SMZE, 2753
    SNCM, 2754
    SS, 2755
    SSLD, 2756
    SSNS, 2757
    SSRLT, 2758
    SSS, 2759
    SUVB, 2760
    SVD, 2761
    TD, 2762
    TDEE, 2763
    technique T, 2764
    TIST, 2765
    TL, 2766
    TMA, 2767
    TMSM, 2768
    TSE, 2769
    TSRM, 2770
    TSS, 2771
    TT, 2772
    TTMI, 2773
    UMCOB, 2774
    UMI, 2775
    UMPIP, 2776
    USR, 2777
    UTMR, 2778
    VFSLS, 2779
    VRI, 2780
    VRILT, 2781
    VRLT, 2782
    VRRB, 2783
    VRS, 2784
    VSLT, 2785
    VSPCV, 2786
    VSPM, 2787
    ZSSM, 2788
    ZVSM, 2789
    ZVU, 2790
ti83
    matrix entry (computation), 2791
    row reduce (computation), 2792
    vector linear combinations (computation), 2793
TI83 (section), 2794
ti86
    matrix entry (computation), 2795
    row reduce (computation), 2796
    transpose of a matrix (computation), 2797
    vector linear combinations (computation), 2798
TI86 (section), 2799
TIS (example), 2800
TIST (theorem), 2801
TIVS (example), 2802
TKAP (example), 2803
TL (theorem), 2804
TLC (example), 2805
TM (definition), 2806
TM (example), 2807
TM (notation), 2808
TM (subsection, section OD), 2809
TM.MMA (computation, section MMA), 2810
TM.SAGE (computation, section SAGE), 2811
TM.TI86 (computation, section TI86), 2812
TMA (theorem), 2813
TMP (example), 2814
TMSM (theorem), 2815
TOV (example), 2816
trace
    definition T, 2817
    linearity
        theorem TL, 2818
    matrix multiplication
        theorem TSRM, 2819
    notation, 2820
    similarity
        theorem TIST, 2821
    sum of eigenvalues
        theorem TSE, 2822
trail mix
    example TMP, 2823
transpose
    matrix scalar multiplication
        theorem TMSM, 2824
    example TM, 2825
    matrix addition
        theorem TMA, 2826
    matrix inverse, 2827, 2828
    notation, 2829
    scalar multiplication, 2830
transpose of a matrix
    mathematica, 2831
    sage, 2832
    ti86, 2833
transpose of a transpose
    theorem TT, 2834
TREM (example), 2835
triangular decomposition
    entry by entry, size 6
        example TDEE6, 2836
    entry by entry
        theorem TDEE, 2837
    size 4
        example TD4, 2838
    solving systems of equations
        example TDSSE, 2839
    theorem TD, 2840
triangular matrix
    inverse
        theorem ITMT, 2841
trivial solution
    system of equations
        definition TSHSE, 2842
TS (definition), 2843
TS (subsection, section S), 2844
TSE (theorem), 2845
TSHSE (definition), 2846
TSM (subsection, section MO), 2847
TSRM (theorem), 2848
TSS (section), 2849
TSS (subsection, section S), 2850
TSS (theorem), 2851
TSVS (definition), 2852
TT (theorem), 2853
TTMI (theorem), 2854
TTS (example), 2855
typical systems, 2 × 2
    example TTS, 2856

U (archetype), 2857
U (technique, section PT), 2858
UM (definition), 2859
UM (subsection, section MINM), 2860
UM3 (example), 2861
UMCOB (theorem), 2862
UMI (theorem), 2863
UMPIP (theorem), 2864
unique solution, 3 × 3
    example US, 2865
    example USR, 2866
uniqueness
    technique U, 2867
unit vectors
    basis
        theorem SUVB, 2868
    definition SUV, 2869
    orthogonal
        example SUVOS, 2870
unitary
    permutation matrix
        example UPM, 2871
    size 3
        example UM3, 2872
unitary matrices
    columns
        theorem CUMOS, 2873
unitary matrix
    inner product
        theorem UMPIP, 2874
UPM (example), 2875
upper triangular matrix
    definition UTM, 2876
US (example), 2877
USR (example), 2878
USR (theorem), 2879
UTM (definition), 2880
UTMR (subsection, section OD), 2881
UTMR (theorem), 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 (chapter), 2972
VS (definition), 2973
VS (section), 2974
VS (subsection, section VS), 2975
VSCV (definition), 2976
VSCV (example), 2977
VSCV (notation), 2978
VSF (example), 2979
VSIM5 (example), 2980
VSIS (example), 2981
VSLT (theorem), 2982
VSM (definition), 2983
VSM (example), 2984
VSM (notation), 2985
VSP (example), 2986
VSP (subsection, section MO), 2987
VSP (subsection, section VO), 2988
VSP (subsection, section VS), 2989
VSPCV (theorem), 2990
VSPM (theorem), 2991
VSPUD (example), 2992
VSS (example), 2993

W (archetype), 2994
WILA (section), 2995

X (archetype), 2996

Z (Property), 2997
ZC (Property), 2998
ZCN (Property), 2999
ZCV (definition), 3000
ZCV (notation), 3001
zero
    complex numbers
        Property ZCN, 3002
    field
        Property ZF, 3003
zero column vector
    definition ZCV, 3004
    notation, 3005
zero matrix
    notation, 3006
zero vector
    column vectors
        Property ZC, 3007
    matrices
        Property ZM, 3008
    unique
        theorem ZVU, 3009
    vectors
        Property Z, 3010
ZF (Property), 3011
ZM (definition), 3012
ZM (notation), 3013
ZM (Property), 3014
ZNDAB (example), 3015
ZSSM (theorem), 3016
ZVSM (theorem), 3017
ZVU (theorem), 3018