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 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 (acronyms, section IVLT), 1477
LT (chapter), 1478
LT (definition), 1479
LT (notation), 1480
LT (section), 1481
LT (subsection, section LT), 1482
LTA (definition), 1483
LTC (definition), 1484
LTC (subsection, section LT), 1485
LTDB (theorem), 1486
LTDB1 (example), 1487
LTDB2 (example), 1488
LTDB3 (example), 1489
LTLC (subsection, section LT), 1490
LTLC (theorem), 1491
LTM (definition), 1492
LTM (example), 1493
LTPM (example), 1494
LTPP (example), 1495
LTR (definition), 1496
LTR (notation), 1497
LTRGE (example), 1498
LTSM (definition), 1499
LTTZZ (theorem), 1500

M (acronyms, section FS), 1501
M (archetype), 1502
M (chapter), 1503
M (definition), 1504
M (notation), 1505
MA (definition), 1506
MA (example), 1507
MA (notation), 1508
MACN (Property), 1509
MAF (Property), 1510
MAP (subsection, section SVD), 1511
mathematica
    gram-schmidt (computation), 1512
    linear solve (computation), 1513
    matrix entry (computation), 1514
    matrix inverse (computation), 1515
    matrix multiplication (computation), 1516
    null space (computation), 1517
    row reduce (computation), 1518
    transpose of a matrix (computation), 1519
    vector form of solutions (computation), 1520
    vector linear combinations (computation), 1521
mathematical language
    technique L, 1522
matrix
    addition
        definition MA, 1523
        notation, 1524
    augmented
        definition AM, 1525
    column space
        definition CSM, 1526
    complex conjugate
        example CCM, 1527
    definition M, 1528
    equality
        definition ME, 1529
        notation, 1530
    example AM, 1531
    identity
        definition IM, 1532
    inverse
        definition MI, 1533
    nonsingular
        definition NM, 1534
    notation, 1535
    of a linear transformation
        theorem MLTCV, 1536
    product
        example PTM, 1537
        example PTMEE, 1538
    product with vector
        definition MVP, 1539
    rectangular, 1540
    row space
        definition RSM, 1541
    scalar multiplication
        definition MSM, 1542
        notation, 1543
    singular, 1544
    square
        definition SQM, 1545
    submatrices
        example SS, 1546
    submatrix
        definition SM, 1547
    symmetric
        definition SYM, 1548
    transpose
        definition TM, 1549
    unitary
        definition UM, 1550
    unitary is invertible
        theorem UMI, 1551
    zero
        definition ZM, 1552
matrix addition
    example MA, 1553
matrix components
    notation, 1554
matrix entry
    mathematica, 1555
    sage, 1556
    ti83, 1557
    ti86, 1558
matrix inverse
    Archetype B, 1559
    computation
        theorem CINM, 1560
    mathematica, 1561
    nonsingular matrix
        theorem NI, 1562
    of a matrix inverse
        theorem MIMI, 1563
    one-sided
        theorem OSIS, 1564
    product
        theorem SS, 1565
    sage, 1566
    scalar multiple
        theorem MISM, 1567
    size 2 matrices
        theorem TTMI, 1568
    transpose
        theorem MIT, 1569
    uniqueness
        theorem MIU, 1570
matrix multiplication
    adjoints
        theorem MMAD, 1571
    associativity
        theorem MMA, 1572
    complex conjugation
        theorem MMCC, 1573
    definition MM, 1574
    distributivity
        theorem MMDAA, 1575
    entry-by-entry
        theorem EMP, 1576
    identity matrix
        theorem MMIM, 1577
    inner product
        theorem MMIP, 1578
    mathematica, 1579
    noncommutative
        example MMNC, 1580
    scalar matrix multiplication
        theorem MMSMM, 1581
    systems of linear equations
        theorem SLEMM, 1582
    transposes
        theorem MMT, 1583
    zero matrix
        theorem MMZM, 1584
matrix product
    as composition of linear transformations
        example MPMR, 1585
matrix representation
    basis of eigenvectors
        example MRBE, 1586
    composition of linear transformations
        theorem MRCLT, 1587
    definition MR, 1588
    invertible
        theorem IMR, 1589
    multiple of a linear transformation
        theorem MRMLT, 1590
    notation, 1591
    restriction to generalized eigenspace
        theorem MRRGE, 1592
    sum of linear transformations
        theorem MRSLT, 1593
    theorem FTMR, 1594
    upper triangular
        theorem UTMR, 1595
matrix representations
    converting with change-of-basis
        example MRCM, 1596
    example OLTTR, 1597
matrix scalar multiplication
    example MSM, 1598
matrix vector space
    dimension
        theorem DM, 1599
matrix-adjoint product
    eigenvalues, eigenvectors
        theorem EEMAP, 1600
matrix-vector product
    example MTV, 1601
    notation, 1602
MBC (example), 1603
MBLT (theorem), 1604
MC (notation), 1605
MCC (subsection, section MO), 1606
MCCN (Property), 1607
MCF (Property), 1608
MCN (definition), 1609
MCN (subsection, section CNO), 1610
MCSM (example), 1611
MCT (theorem), 1612
MD (chapter), 1613
ME (definition), 1614
ME (notation), 1615
ME (subsection, section PEE), 1616
ME (technique, section PT), 1617
ME (theorem), 1618
ME.MMA (computation, section MMA), 1619
ME.SAGE (computation, section SAGE), 1620
ME.TI83 (computation, section TI83), 1621
ME.TI86 (computation, section TI86), 1622
MEASM (subsection, section MO), 1623
MFLT (example), 1624
MI (definition), 1625
MI (example), 1626
MI (notation), 1627
MI.MMA (computation, section MMA), 1628
MI.SAGE (computation, section SAGE), 1629
MICN (Property), 1630
MIF (Property), 1631
MIMI (theorem), 1632
MINM (section), 1633
MISLE (section), 1634
MISM (theorem), 1635
MIT (theorem), 1636
MIU (theorem), 1637
MIVS (example), 1638
MLT (subsection, section LT), 1639
MLTCV (theorem), 1640
MLTLT (theorem), 1641
MM (definition), 1642
MM (section), 1643
MM (subsection, section MM), 1644
MM.MMA (computation, section MMA), 1645
MMA (section), 1646
MMA (theorem), 1647
MMAD (theorem), 1648
MMCC (theorem), 1649
MMDAA (theorem), 1650
MMEE (subsection, section MM), 1651
MMIM (theorem), 1652
MMIP (theorem), 1653
MMNC (example), 1654
MMSMM (theorem), 1655
MMT (theorem), 1656
MMZM (theorem), 1657
MNEM (theorem), 1658
MNSLE (example), 1659
MO (section), 1660
MOLT (example), 1661
more variables than equations
    example OSGMD, 1662
    theorem CMVEI, 1663
MPMR (example), 1664
MR (definition), 1665
MR (notation), 1666
MR (section), 1667
MRBE (example), 1668
MRCB (theorem), 1669
MRCLT (diagram), 1670
MRCLT (theorem), 1671
MRCM (example), 1672
MRLS (definition), 1673
MRLS (notation), 1674
MRMLT (theorem), 1675
MRRGE (theorem), 1676
MRS (subsection, section CB), 1677
MRSLT (theorem), 1678
MSCN (example), 1679
MSM (definition), 1680
MSM (example), 1681
MSM (notation), 1682
MTV (example), 1683
multiplicative associativity
    complex numbers
        Property MACN, 1684
multiplicative closure
    complex numbers
        Property MCCN, 1685
    field
        Property MCF, 1686
multiplicative commutativity
    complex numbers
        Property CMCN, 1687
multiplicative inverse
    complex numbers
        Property MICN, 1688
MVNSE (subsection, section RREF), 1689
MVP (definition), 1690
MVP (notation), 1691
MVP (subsection, section MM), 1692
MVSLD (theorem), 1693
MWIAA (example), 1694

N (archetype), 1695
N (subsection, section O), 1696
N (technique, section PT), 1697
NDMS4 (example), 1698
negation of statements
    technique N, 1699
NEM (theorem), 1700
NI (theorem), 1701
NIAO (example), 1702
NIAQ (example), 1703
NIAQR (example), 1704
NIDAU (example), 1705
nilpotent
    linear transformation
        definition NLT, 1706
NILT (diagram), 1707
NJB (theorem), 1708
NJB5 (example), 1709
NKAO (example), 1710
NLT (definition), 1711
NLT (example), 1712
NLT (section), 1713
NLT (subsection, section NLT), 1714
NLTFO (subsection, section LT), 1715
NM (definition), 1716
NM (example), 1717
NM (section), 1718
NM (subsection, section NM), 1719
NM (subsection, section OD), 1720
NM62 (example), 1721
NM64 (example), 1722
NM83 (example), 1723
NME1 (theorem), 1724
NME2 (theorem), 1725
NME3 (theorem), 1726
NME4 (theorem), 1727
NME5 (theorem), 1728
NME6 (theorem), 1729
NME7 (theorem), 1730
NME8 (theorem), 1731
NME9 (theorem), 1732
NMI (subsection, section MINM), 1733
NMLIC (theorem), 1734
NMPEM (theorem), 1735
NMRRI (theorem), 1736
NMTNS (theorem), 1737
NMUS (theorem), 1738
NOILT (theorem), 1739
NOLT (definition), 1740
NOLT (notation), 1741
NOM (definition), 1742
NOM (notation), 1743
nonsingular
    columns as basis
        theorem CNMB, 1744
nonsingular matrices
    linearly independent columns
        theorem NMLIC, 1745
nonsingular matrix
    Archetype B
        example NM, 1746
    column space, 1747
    elementary matrices
        theorem NMPEM, 1748
    equivalences
        theorem NME1, 1749
        theorem NME2, 1750
        theorem NME3, 1751
        theorem NME4, 1752
        theorem NME5, 1753
        theorem NME6, 1754
        theorem NME7, 1755
        theorem NME8, 1756
        theorem NME9, 1757
    matrix inverse, 1758
    null space
        example NSNM, 1759
    nullity, 1760
    product of nonsingular matrices
        theorem NPNT, 1761
    rank
        theorem RNNM, 1762
    row-reduced
        theorem NMRRI, 1763
    trivial null space
        theorem NMTNS, 1764
    unique solutions
        theorem NMUS, 1765
nonsingular matrix, row-reduced
    example NSR, 1766
norm
    example CNSV, 1767
    inner product, 1768
    notation, 1769
normal matrix
    definition NRML, 1770
    example ANM, 1771
    orthonormal basis, 1772
notation
    A, 1773
    AM, 1774
    AME, 1775
    C, 1776
    CCCV, 1777
    CCM, 1778
    CCN, 1779
    CNA, 1780
    CNE, 1781
    CNM, 1782
    CSM, 1783
    CV, 1784
    CVA, 1785
    CVC, 1786
    CVE, 1787
    CVSM, 1788
    D, 1789
    DM, 1790
    DS, 1791
    ELEM, 1792
    ES, 1793
    GES, 1794
    GME, 1795
    HI, 1796
    HID, 1797
    HP, 1798
    IE, 1799
    IM, 1800
    IP, 1801
    JB, 1802
    KLT, 1803
    LNS, 1804
    LT, 1805
    LTR, 1806
    M, 1807
    MA, 1808
    MC, 1809
    ME, 1810
    MI, 1811
    MR, 1812
    MRLS, 1813
    MSM, 1814
    MVP, 1815
    NOLT, 1816
    NOM, 1817
    NSM, 1818
    NV, 1819
    RLT, 1820
    RO, 1821
    ROLT, 1822
    ROM, 1823
    RREFA, 1824
    RSM, 1825
    SC, 1826
    SE, 1827
    SETM, 1828
    SI, 1829
    SM, 1830
    SRM, 1831
    SSET, 1832
    SSV, 1833
    SU, 1834
    SUV, 1835
    T, 1836
    TM, 1837
    VR, 1838
    VSCV, 1839
    VSM, 1840
    ZCV, 1841
    ZM, 1842
notation for a linear system
    example NSE, 1843
NPNT (theorem), 1844
NRFO (subsection, section MR), 1845
NRML (definition), 1846
NRREF (example), 1847
NS.MMA (computation, section MMA), 1848
NSAO (example), 1849
NSAQ (example), 1850
NSAQR (example), 1851
NSC2A (example), 1852
NSC2S (example), 1853
NSC2Z (example), 1854
NSDAT (example), 1855
NSDS (example), 1856
NSE (example), 1857
NSEAI (example), 1858
NSLE (example), 1859
NSLIL (example), 1860
NSM (definition), 1861
NSM (notation), 1862
NSM (subsection, section HSE), 1863
NSMS (theorem), 1864
NSNM (example), 1865
NSNM (subsection, section NM), 1866
NSR (example), 1867
NSS (example), 1868
NSSLI (subsection, section LI), 1869
Null space
    as a span
        example NSDS, 1870
null space
    Archetype I
        example NSEAI, 1871
    basis
        theorem BNS, 1872
    computation
        example CNS1, 1873
        example CNS2, 1874
    isomorphic to kernel, 1875
    linearly independent basis
        example LINSB, 1876
    mathematica, 1877
    matrix
        definition NSM, 1878
    nonsingular matrix, 1879
    notation, 1880
    singular matrix, 1881
    spanning set
        example SSNS, 1882
        theorem SSNS, 1883
    subspace
        theorem NSMS, 1884
null space span, linearly independent
    Archetype L
        example NSLIL, 1885
nullity
    computing, 1886
    injective linear transformation
        theorem NOILT, 1887
    linear transformation
        definition NOLT, 1888
    matrix, 1889
        definition NOM, 1890
    notation, 1891, 1892
    square matrix, 1893
NV (definition), 1894
NV (notation), 1895
NVM (theorem), 1896

O (archetype), 1897
O (Property), 1898
O (section), 1899
OBC (subsection, section B), 1900
OBNM (theorem), 1901
OBUTR (theorem), 1902
OC (Property), 1903
OCN (Property), 1904
OD (section), 1905
OD (subsection, section OD), 1906
OD (theorem), 1907
OF (Property), 1908
OLTTR (example), 1909
OM (Property), 1910
one
    column vectors
        Property OC, 1911
    complex numbers
        Property OCN, 1912
    field
        Property OF, 1913
    matrices
        Property OM, 1914
    vectors
        Property O, 1915
ONFV (example), 1916
ONS (definition), 1917
ONTV (example), 1918
orthogonal
    linear independence
        theorem OSLI, 1919
    set
        example AOS, 1920
    set of vectors
        definition OSV, 1921
    vector pairs
        definition OV, 1922
orthogonal vectors
    example TOV, 1923
orthonormal
    definition ONS, 1924
    matrix columns
        example OSMC, 1925
orthonormal basis
    normal matrix
        theorem OBNM, 1926
orthonormal diagonalization
    theorem OD, 1927
orthonormal set
    four vectors
        example ONFV, 1928
    three vectors
        example ONTV, 1929
OSGMD (example), 1930
OSIS (theorem), 1931
OSLI (theorem), 1932
OSMC (example), 1933
OSV (definition), 1934
OV (definition), 1935
OV (subsection, section O), 1936

P (appendix), 1937
P (archetype), 1938
P (technique, section PT), 1939
particular solutions
    example PSHS, 1940
PCNA (theorem), 1941
PCVS (example), 1942
PD (section), 1943
PDM (section), 1944
PDM (theorem), 1945
PEE (section), 1946
PEEF (theorem), 1947
PI (definition), 1948
PI (subsection, section LT), 1949
PI (technique, section PT), 1950
PIP (theorem), 1951
PM (example), 1952
PM (subsection, section EE), 1953
PMI (subsection, section MISLE), 1954
PMM (subsection, section MM), 1955
PMR (subsection, section MR), 1956
PNLT (subsection, section NLT), 1957
POD (section), 1958
polar decomposition
    theorem PDM, 1959
polynomial
    of a matrix
        example PM, 1960
polynomial vector space
    dimension
        theorem DP, 1961
positive semi-definite
    creating
        theorem CPSM, 1962
positive semi-definite matrix
    definition PSM, 1963
    eigenvalues
        theorem EPSM, 1964
practice
    technique P, 1965
pre-image
    definition PI, 1966
    kernel
        theorem KPI, 1967
pre-images
    example SPIAS, 1968
principal axis theorem, 1969
product of triangular matrices
    theorem PTMT, 1970
Property
    AA, 1971
    AAC, 1972
    AACN, 1973
    AAF, 1974
    AAM, 1975
    AC, 1976
    ACC, 1977
    ACCN, 1978
    ACF, 1979
    ACM, 1980
    AI, 1981
    AIC, 1982
    AICN, 1983
    AIF, 1984
    AIM, 1985
    C, 1986
    CACN, 1987
    CAF, 1988
    CC, 1989
    CM, 1990
    CMCN, 1991
    CMF, 1992
    DCN, 1993
    DF, 1994
    DMAM, 1995
    DSA, 1996
    DSAC, 1997
    DSAM, 1998
    DVA, 1999
    DVAC, 2000
    MACN, 2001
    MAF, 2002
    MCCN, 2003
    MCF, 2004
    MICN, 2005
    MIF, 2006
    O, 2007
    OC, 2008
    OCN, 2009
    OF, 2010
    OM, 2011
    SC, 2012
    SCC, 2013
    SCM, 2014
    SMA, 2015
    SMAC, 2016
    SMAM, 2017
    Z, 2018
    ZC, 2019
    ZCN, 2020
    ZF, 2021
    ZM, 2022
PSHS (example), 2023
PSHS (subsection, section LC), 2024
PSM (definition), 2025
PSM (section), 2026
PSM (subsection, section PSM), 2027
PSM (subsection, section SD), 2028
PSMSR (theorem), 2029
PSPHS (theorem), 2030
PSS (subsection, section SSLE), 2031
PSSD (theorem), 2032
PSSLS (theorem), 2033
PT (section), 2034
PTFP (example), 2035
PTM (example), 2036
PTMEE (example), 2037
PTMT (theorem), 2038

Q (archetype), 2039

R (acronyms, section JCF), 2040
R (archetype), 2041
R (chapter), 2042
R.SAGE (computation, section SAGE), 2043
range
    full
        example FRAN, 2044
    isomorphic to column space
        theorem RCSI, 2045
    linear transformation
        example RAO, 2046
    notation, 2047
    of a linear transformation
        definition RLT, 2048
    pre-image
        theorem RPI, 2049
    subspace
        theorem RLTS, 2050
    surjective linear transformation
        theorem RSLT, 2051
    via matrix representation
        example RVMR, 2052
rank
    computing
        theorem CRN, 2053
    linear transformation
        definition ROLT, 2054
    matrix
        definition ROM, 2055
        example RNM, 2056
    notation, 2057, 2058
    of transpose
        example RRTI, 2059
    square matrix
        example RNSM, 2060
    surjective linear transformation
        theorem ROSLT, 2061
    transpose
        theorem RMRT, 2062
rank one decomposition
    size 2
        example ROD2, 2063
    size 4
        example ROD4, 2064
    theorem ROD, 2065
rank+nullity
    theorem RPNC, 2066
RAO (example), 2067
RCLS (theorem), 2068
RCSI (theorem), 2069
RD (subsection, section VS), 2070
RDS (theorem), 2071
READ (subsection, section B), 2072
READ (subsection, section CB), 2073
READ (subsection, section CRS), 2074
READ (subsection, section D), 2075
READ (subsection, section DM), 2076
READ (subsection, section EE), 2077
READ (subsection, section FS), 2078
READ (subsection, section HSE), 2079
READ (subsection, section ILT), 2080
READ (subsection, section IVLT), 2081
READ (subsection, section LC), 2082
READ (subsection, section LDS), 2083
READ (subsection, section LI), 2084
READ (subsection, section LISS), 2085
READ (subsection, section LT), 2086
READ (subsection, section MINM), 2087
READ (subsection, section MISLE), 2088
READ (subsection, section MM), 2089
READ (subsection, section MO), 2090
READ (subsection, section MR), 2091
READ (subsection, section NM), 2092
READ (subsection, section O), 2093
READ (subsection, section PD), 2094
READ (subsection, section PDM), 2095
READ (subsection, section PEE), 2096
READ (subsection, section RREF), 2097
READ (subsection, section S), 2098
READ (subsection, section SD), 2099
READ (subsection, section SLT), 2100
READ (subsection, section SS), 2101
READ (subsection, section SSLE), 2102
READ (subsection, section TSS), 2103
READ (subsection, section VO), 2104
READ (subsection, section VR), 2105
READ (subsection, section VS), 2106
READ (subsection, section WILA), 2107
reduced row-echelon form
    analysis
        notation, 2108
    definition RREF, 2109
    example NRREF, 2110
    example RREF, 2111
    extended
        definition EEF, 2112
    notation
        example RREFN, 2113
    unique
        theorem RREFU, 2114
reducing a span
    example RSC5, 2115
relation of linear dependence
    definition RLD, 2116
    definition RLDCV, 2117
REM (definition), 2118
REMEF (theorem), 2119
REMES (theorem), 2120
REMRS (theorem), 2121
RES (example), 2122
RGEN (theorem), 2123
rings
    sage, 2124
RLD (definition), 2125
RLDCV (definition), 2126
RLT (definition), 2127
RLT (notation), 2128
RLT (subsection, section IS), 2129
RLT (subsection, section SLT), 2130
RLTS (theorem), 2131
RMRT (theorem), 2132
RNLT (subsection, section IVLT), 2133
RNM (example), 2134
RNM (subsection, section D), 2135
RNNM (subsection, section D), 2136
RNNM (theorem), 2137
RNSM (example), 2138
RO (definition), 2139
RO (notation), 2140
RO (subsection, section RREF), 2141
ROD (section), 2142
ROD (theorem), 2143
ROD2 (example), 2144
ROD4 (example), 2145
ROLT (definition), 2146
ROLT (notation), 2147
ROM (definition), 2148
ROM (notation), 2149
ROSLT (theorem), 2150
row operations
    definition RO, 2151
    elementary matrices, 2152, 2153
    notation, 2154
row reduce
    mathematica, 2155
    sage, 2156
    ti83, 2157
    ti86, 2158
row space
    Archetype I
        example RSAI, 2159
    as column space, 2160
    basis
        example RSB, 2161
        theorem BRS, 2162
    matrix, 2163
    notation, 2164
    row-equivalent matrices
        theorem REMRS, 2165
    subspace
        theorem RSMS, 2166
row-equivalent matrices
    definition REM, 2167
    example TREM, 2168
    row space, 2169
    row spaces
        example RSREM, 2170
    theorem REMES, 2171
row-reduce
    the verb
        definition RR, 2172
row-reduced matrices
    theorem REMEF, 2173
RPI (theorem), 2174
RPNC (theorem), 2175
RPNDD (theorem), 2176
RR (definition), 2177
RR.MMA (computation, section MMA), 2178
RR.SAGE (computation, section SAGE), 2179
RR.TI83 (computation, section TI83), 2180
RR.TI86 (computation, section TI86), 2181
RREF (definition), 2182
RREF (example), 2183
RREF (section), 2184
RREF (subsection, section RREF), 2185
RREFA (notation), 2186
RREFN (example), 2187
RREFU (theorem), 2188
RRTI (example), 2189
RS (example), 2190
RSAI (example), 2191
RSB (example), 2192
RSC4 (example), 2193
RSC5 (example), 2194
RSLT (theorem), 2195
RSM (definition), 2196
RSM (notation), 2197
RSM (subsection, section CRS), 2198
RSMS (theorem), 2199
RSNS (example), 2200
RSREM (example), 2201
RT (subsection, section PD), 2202
RVMR (example), 2203

S (archetype), 2204
S (definition), 2205
S (example), 2206
S (section), 2207
SAA (example), 2208
SAB (example), 2209
SABMI (example), 2210
SAE (example), 2211
sage
    eigenspaces (computation), 2212
    linear solve (computation), 2213
    matrix entry (computation), 2214
    matrix inverse (computation), 2215
    rings (computation), 2216
    row reduce (computation), 2217
    transpose of a matrix (computation), 2218
    vector linear combinations (computation), 2219
SAGE (section), 2220
SAN (example), 2221
SAR (example), 2222
SAS (section), 2223
SAV (example), 2224
SC (definition), 2225
SC (example), 2226
SC (notation), 2227
SC (Property), 2228
SC (subsection, section S), 2229
SC (subsection, section SET), 2230
SC3 (example), 2231
SCAA (example), 2232
SCAB (example), 2233
SCAD (example), 2234
scalar closure
    column vectors
        Property SCC, 2235
    matrices
        Property SCM, 2236
    vectors
        Property SC, 2237
scalar multiple
    matrix inverse, 2238
scalar multiplication
    zero scalar
        theorem ZSSM, 2239
    zero vector
        theorem ZVSM, 2240
    zero vector result
        theorem SMEZV, 2241
scalar multiplication associativity
    column vectors
        Property SMAC, 2242
    matrices
        Property SMAM, 2243
    vectors
        Property SMA, 2244
SCB (theorem), 2245
SCC (Property), 2246
SCM (Property), 2247
SD (section), 2248
SDS (example), 2249
SE (definition), 2250
SE (notation), 2251
secret sharing
    6 ways
        example SS6W, 2252
SEE (example), 2253
SEEF (example), 2254
SER (theorem), 2255
set
    cardinality
        definition C, 2256
        example CS, 2257
        notation, 2258
    complement
        definition SC, 2259
        example SC, 2260
        notation, 2261
    definition SET, 2262
    empty
        definition ES, 2263
    equality
        definition SE, 2264
        notation, 2265
    intersection
        definition SI, 2266
        example SI, 2267
        notation, 2268
    membership
        example SETM, 2269
        notation, 2270
    size, 2271
    subset, 2272
    union
        definition SU, 2273
        example SU, 2274
        notation, 2275
SET (definition), 2276
SET (section), 2277
SETM (example), 2278
SETM (notation), 2279
shoes, 2280
SHS (subsection, section HSE), 2281
SI (definition), 2282
SI (example), 2283
SI (notation), 2284
SI (subsection, section IVLT), 2285
SIM (definition), 2286
similar matrices
    equal eigenvalues
        example EENS, 2287
    eual eigenvalues
        theorem SMEE, 2288
    example SMS3, 2289
    example SMS5, 2290
similarity
    definition SIM, 2291
    equivalence relation
        theorem SER, 2292
singular matrix
    Archetype A
        example S, 2293
    null space
        example NSS, 2294
singular matrix, row-reduced
    example SRR, 2295
singular value decomposition
    theorem SVD, 2296
singular values
    definition SV, 2297
SLE (acronyms, section NM), 2298
SLE (chapter), 2299
SLE (definition), 2300
SLE (subsection, section SSLE), 2301
SLELT (subsection, section IVLT), 2302
SLEMM (theorem), 2303
SLSLC (theorem), 2304
SLT (definition), 2305
SLT (section), 2306
SLTB (theorem), 2307
SLTD (subsection, section SLT), 2308
SLTD (theorem), 2309
SLTLT (theorem), 2310
SM (definition), 2311
SM (notation), 2312
SM (subsection, section SD), 2313
SM2Z7 (example), 2314
SM32 (example), 2315
SMA (Property), 2316
SMAC (Property), 2317
SMAM (Property), 2318
SMEE (theorem), 2319
SMEZV (theorem), 2320
SMLT (example), 2321
SMS (theorem), 2322
SMS3 (example), 2323
SMS5 (example), 2324
SMZD (theorem), 2325
SMZE (theorem), 2326
SNCM (theorem), 2327
SO (subsection, section SET), 2328
socks, 2329
SOL (subsection, section B), 2330
SOL (subsection, section CB), 2331
SOL (subsection, section CRS), 2332
SOL (subsection, section D), 2333
SOL (subsection, section DM), 2334
SOL (subsection, section EE), 2335
SOL (subsection, section F), 2336
SOL (subsection, section FS), 2337
SOL (subsection, section HSE), 2338
SOL (subsection, section ILT), 2339
SOL (subsection, section IVLT), 2340
SOL (subsection, section LC), 2341
SOL (subsection, section LDS), 2342
SOL (subsection, section LI), 2343
SOL (subsection, section LISS), 2344
SOL (subsection, section LT), 2345
SOL (subsection, section MINM), 2346
SOL (subsection, section MISLE), 2347
SOL (subsection, section MM), 2348
SOL (subsection, section MO), 2349
SOL (subsection, section MR), 2350
SOL (subsection, section NM), 2351
SOL (subsection, section O), 2352
SOL (subsection, section PD), 2353
SOL (subsection, section PDM), 2354
SOL (subsection, section PEE), 2355
SOL (subsection, section RREF), 2356
SOL (subsection, section S), 2357
SOL (subsection, section SD), 2358
SOL (subsection, section SLT), 2359
SOL (subsection, section SS), 2360
SOL (subsection, section SSLE), 2361
SOL (subsection, section T), 2362
SOL (subsection, section TSS), 2363
SOL (subsection, section VO), 2364
SOL (subsection, section VR), 2365
SOL (subsection, section VS), 2366
SOL (subsection, section WILA), 2367
solution set
    Archetype A
        example SAA, 2368
    archetype E
        example SAE, 2369
    theorem PSPHS, 2370
solution sets
    possibilities
        theorem PSSLS, 2371
solution vector
    definition SOLV, 2372
SOLV (definition), 2373
solving homogeneous system
    Archetype A
        example HISAA, 2374
    Archetype B
        example HUSAB, 2375
    Archetype D
        example HISAD, 2376
solving nonlinear equations
    example STNE, 2377
SP4 (example), 2378
span
    basic
        example ABS, 2379
    basis
        theorem BS, 2380
    definition SS, 2381
    definition SSCV, 2382
    improved
        example IAS, 2383
    notation, 2384
    reducing
        example RSC4, 2385
    reduction
        example RS, 2386
    removing vectors
        example COV, 2387
    reworking elements
        example RES, 2388
    set of polynomials
        example SSP, 2389
    subspace
        theorem SSS, 2390
span of columns
    Archetype A
        example SCAA, 2391
    Archetype B
        example SCAB, 2392
    Archetype D
        example SCAD, 2393
spanning set
    crazy vector space
        example SSC, 2394
    definition TSVS, 2395
    matrices
        example SSM22, 2396
    more vectors
        theorem SSLD, 2397
    polynomials
        example SSP4, 2398
SPIAS (example), 2399
SQM (definition), 2400
square root
    eigenvalues, eigenspaces
        theorem EESR, 2401
    matrix
        definition SRM, 2402
        notation, 2403
    positive semi-definite matrix
        theorem PSMSR, 2404
    unique
        theorem USR, 2405
SR (section), 2406
SRM (definition), 2407
SRM (notation), 2408
SRM (subsection, section SR), 2409
SRR (example), 2410
SS (definition), 2411
SS (example), 2412
SS (section), 2413
SS (subsection, section LISS), 2414
SS (theorem), 2415
SS6W (example), 2416
SSC (example), 2417
SSCV (definition), 2418
SSET (definition), 2419
SSET (example), 2420
SSET (notation), 2421
SSLD (theorem), 2422
SSLE (section), 2423
SSM22 (example), 2424
SSNS (example), 2425
SSNS (subsection, section SS), 2426
SSNS (theorem), 2427
SSP (example), 2428
SSP4 (example), 2429
SSRLT (theorem), 2430
SSS (theorem), 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 (acronyms, section O), 2883
V (archetype), 2884
V (chapter), 2885
VA (example), 2886
Vandermonde matrix
    definition VM, 2887
vandermonde matrix
    determinant
        theorem DVM, 2888
    nonsingular
        theorem NVM, 2889
    size 4
        example VM4, 2890
VEASM (subsection, section VO), 2891
vector
    addition
        definition CVA, 2892
    column
        definition CV, 2893
    equality
        definition CVE, 2894
        notation, 2895
    inner product
        definition IP, 2896
    norm
        definition NV, 2897
    notation, 2898
    of constants
        definition VOC, 2899
    product with matrix, 2900, 2901
    scalar multiplication
        definition CVSM, 2902
vector addition
    example VA, 2903
vector component
    notation, 2904
vector form of solutions
    Archetype D
        example VFSAD, 2905
    Archetype I
        example VFSAI, 2906
    Archetype L
        example VFSAL, 2907
    example VFS, 2908
    mathematica, 2909
    theorem VFSLS, 2910
vector linear combinations
    mathematica, 2911
    sage, 2912
    ti83, 2913
    ti86, 2914
vector representation
    example AVR, 2915
    example VRC4, 2916
    injective
        theorem VRI, 2917
    invertible
        theorem VRILT, 2918
    linear transformation
        definition VR, 2919
        notation, 2920
        theorem VRLT, 2921
    surjective
        theorem VRS, 2922
    theorem VRRB, 2923
vector representations
    polynomials
        example VRP2, 2924
vector scalar multiplication
    example CVSM, 2925
vector space
    characterization
        theorem CFDVS, 2926
    column vectors
        definition VSCV, 2927
    definition VS, 2928
    infinite dimension
        example VSPUD, 2929
    linear transformations
        theorem VSLT, 2930
    over integers mod 5
        example VSIM5, 2931
vector space of column vectors
    notation, 2932
vector space of functions
    example VSF, 2933
vector space of infinite sequences
    example VSIS, 2934
vector space of matrices
    definition VSM, 2935
    example VSM, 2936
    notation, 2937
vector space of polynomials
    example VSP, 2938
vector space properties
    column vectors
        theorem VSPCV, 2939
    matrices
        theorem VSPM, 2940
vector space, crazy
    example CVS, 2941
vector space, singleton
    example VSS, 2942
vector spaces
    isomorphic
        definition IVS, 2943
        theorem IFDVS, 2944
VESE (example), 2945
VFS (example), 2946
VFSAD (example), 2947
VFSAI (example), 2948
VFSAL (example), 2949
VFSLS (theorem), 2950
VFSS (subsection, section LC), 2951
VFSS.MMA (computation, section MMA), 2952
VLC.MMA (computation, section MMA), 2953
VLC.SAGE (computation, section SAGE), 2954
VLC.TI83 (computation, section TI83), 2955
VLC.TI86 (computation, section TI86), 2956
VM (definition), 2957
VM (section), 2958
VM4 (example), 2959
VO (section), 2960
VOC (definition), 2961
VR (definition), 2962
VR (notation), 2963
VR (section), 2964
VR (subsection, section LISS), 2965
VRC4 (example), 2966
VRI (theorem), 2967
VRILT (theorem), 2968
VRLT (theorem), 2969
VRP2 (example), 2970
VRRB (theorem), 2971
VRS (theorem), 2972
VS (acronyms, section PD), 2973
VS (chapter), 2974
VS (definition), 2975
VS (section), 2976
VS (subsection, section VS), 2977
VSCV (definition), 2978
VSCV (example), 2979
VSCV (notation), 2980
VSF (example), 2981
VSIM5 (example), 2982
VSIS (example), 2983
VSLT (theorem), 2984
VSM (definition), 2985
VSM (example), 2986
VSM (notation), 2987
VSP (example), 2988
VSP (subsection, section MO), 2989
VSP (subsection, section VO), 2990
VSP (subsection, section VS), 2991
VSPCV (theorem), 2992
VSPM (theorem), 2993
VSPUD (example), 2994
VSS (example), 2995

W (archetype), 2996
WILA (section), 2997

X (archetype), 2998

Z (Property), 2999
ZC (Property), 3000
ZCN (Property), 3001
ZCV (definition), 3002
ZCV (notation), 3003
zero
    complex numbers
        Property ZCN, 3004
    field
        Property ZF, 3005
zero column vector
    definition ZCV, 3006
    notation, 3007
zero matrix
    notation, 3008
zero vector
    column vectors
        Property ZC, 3009
    matrices
        Property ZM, 3010
    unique
        theorem ZVU, 3011
    vectors
        Property Z, 3012
ZF (Property), 3013
ZM (definition), 3014
ZM (notation), 3015
ZM (Property), 3016
ZNDAB (example), 3017
ZSSM (theorem), 3018
ZVSM (theorem), 3019
ZVU (theorem), 3020