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 commutativity
    complex numbers
        Property CACN, 26
additive inverse
    complex numbers
        Property AICN, 27
    from scalar multiplication
        theorem AISM, 28
additive inverses
    column vectors
        Property AIC, 29
    matrices
        Property AIM, 30
    unique
        theorem AIU, 31
    vectors
        Property AI, 32
addtive closure
    column vectors
        Property ACC, 33
    complex numbers
        Property ACCN, 34
    field
        Property ACF, 35
    matrices
        Property ACM, 36
    vectors
        Property AC, 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
    RSC5, 952
    RSNS, 953
    RSREM, 954
    RSSC4, 955
    RVMR, 956
    S, 957
    SAA, 958
    SAB, 959
    SABMI, 960
    SAE, 961
    SAN, 962
    SAR, 963
    SAV, 964
    SC, 965
    SC3, 966
    SCAA, 967
    SCAB, 968
    SCAD, 969
    SDS, 970
    SEE, 971
    SEEF, 972
    SETM, 973
    SI, 974
    SM2Z7, 975
    SM32, 976
    SMLT, 977
    SMS3, 978
    SMS5, 979
    SP4, 980
    SPIAS, 981
    SRR, 982
    SS, 983
    SS6W, 984
    SSC, 985
    SSET, 986
    SSM22, 987
    SSNS, 988
    SSP, 989
    SSP4, 990
    STLT, 991
    STNE, 992
    SU, 993
    SUVOS, 994
    SVP4, 995
    SYM, 996
    TCSD, 997
    TD4, 998
    TDEE6, 999
    TDSSE, 1000
    TIS, 1001
    TIVS, 1002
    TKAP, 1003
    TLC, 1004
    TM, 1005
    TMP, 1006
    TOV, 1007
    TREM, 1008
    TTS, 1009
    UM3, 1010
    UPM, 1011
    US, 1012
    USR, 1013
    VA, 1014
    VESE, 1015
    VFS, 1016
    VFSAD, 1017
    VFSAI, 1018
    VFSAL, 1019
    VM4, 1020
    VRC4, 1021
    VRP2, 1022
    VSCV, 1023
    VSF, 1024
    VSIM5, 1025
    VSIS, 1026
    VSM, 1027
    VSP, 1028
    VSPUD, 1029
    VSS, 1030
    ZNDAB, 1031
EXC (subsection, section B), 1032
EXC (subsection, section CB), 1033
EXC (subsection, section CF), 1034
EXC (subsection, section CRS), 1035
EXC (subsection, section D), 1036
EXC (subsection, section DM), 1037
EXC (subsection, section EE), 1038
EXC (subsection, section F), 1039
EXC (subsection, section FS), 1040
EXC (subsection, section HP), 1041
EXC (subsection, section HSE), 1042
EXC (subsection, section ILT), 1043
EXC (subsection, section IVLT), 1044
EXC (subsection, section LC), 1045
EXC (subsection, section LDS), 1046
EXC (subsection, section LI), 1047
EXC (subsection, section LISS), 1048
EXC (subsection, section LT), 1049
EXC (subsection, section MINM), 1050
EXC (subsection, section MISLE), 1051
EXC (subsection, section MM), 1052
EXC (subsection, section MO), 1053
EXC (subsection, section MR), 1054
EXC (subsection, section NM), 1055
EXC (subsection, section O), 1056
EXC (subsection, section PD), 1057
EXC (subsection, section PDM), 1058
EXC (subsection, section PEE), 1059
EXC (subsection, section PSM), 1060
EXC (subsection, section RREF), 1061
EXC (subsection, section S), 1062
EXC (subsection, section SD), 1063
EXC (subsection, section SLT), 1064
EXC (subsection, section SS), 1065
EXC (subsection, section SSLE), 1066
EXC (subsection, section T), 1067
EXC (subsection, section TSS), 1068
EXC (subsection, section VO), 1069
EXC (subsection, section VR), 1070
EXC (subsection, section VS), 1071
EXC (subsection, section WILA), 1072
extended echelon form
    submatrices
        example SEEF, 1073
extended reduced row-echelon form
    properties
        theorem PEEF, 1074

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

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

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

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

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

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

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

M (acronyms, section FS), 1498
M (archetype), 1499
M (chapter), 1500
M (definition), 1501
M (notation), 1502
MA (definition), 1503
MA (example), 1504
MA (notation), 1505
MACN (Property), 1506
MAF (Property), 1507
MAP (subsection, section SVD), 1508
mathematica
    gram-schmidt (computation), 1509
    linear solve (computation), 1510
    matrix entry (computation), 1511
    matrix inverse (computation), 1512
    matrix multiplication (computation), 1513
    null space (computation), 1514
    row reduce (computation), 1515
    transpose of a matrix (computation), 1516
    vector form of solutions (computation), 1517
    vector linear combinations (computation), 1518
mathematical language
    technique L, 1519
matrix
    addition
        definition MA, 1520
        notation, 1521
    augmented
        definition AM, 1522
    column space
        definition CSM, 1523
    complex conjugate
        example CCM, 1524
    definition M, 1525
    equality
        definition ME, 1526
        notation, 1527
    example AM, 1528
    identity
        definition IM, 1529
    inverse
        definition MI, 1530
    nonsingular
        definition NM, 1531
    notation, 1532
    of a linear transformation
        theorem MLTCV, 1533
    product
        example PTM, 1534
        example PTMEE, 1535
    product with vector
        definition MVP, 1536
    rectangular, 1537
    row space
        definition RSM, 1538
    scalar multiplication
        definition MSM, 1539
        notation, 1540
    singular, 1541
    square
        definition SQM, 1542
    submatrices
        example SS, 1543
    submatrix
        definition SM, 1544
    symmetric
        definition SYM, 1545
    transpose
        definition TM, 1546
    unitary
        definition UM, 1547
    unitary is invertible
        theorem UMI, 1548
    zero
        definition ZM, 1549
matrix addition
    example MA, 1550
matrix components
    notation, 1551
matrix entry
    mathematica, 1552
    sage, 1553
    ti83, 1554
    ti86, 1555
matrix inverse
    Archetype B, 1556
    computation
        theorem CINM, 1557
    mathematica, 1558
    nonsingular matrix
        theorem NI, 1559
    of a matrix inverse
        theorem MIMI, 1560
    one-sided
        theorem OSIS, 1561
    product
        theorem SS, 1562
    sage, 1563
    scalar multiple
        theorem MISM, 1564
    size 2 matrices
        theorem TTMI, 1565
    transpose
        theorem MIT, 1566
    uniqueness
        theorem MIU, 1567
matrix multiplication
    adjoints
        theorem MMAD, 1568
    associativity
        theorem MMA, 1569
    complex conjugation
        theorem MMCC, 1570
    definition MM, 1571
    distributivity
        theorem MMDAA, 1572
    entry-by-entry
        theorem EMP, 1573
    identity matrix
        theorem MMIM, 1574
    inner product
        theorem MMIP, 1575
    mathematica, 1576
    noncommutative
        example MMNC, 1577
    scalar matrix multiplication
        theorem MMSMM, 1578
    systems of linear equations
        theorem SLEMM, 1579
    transposes
        theorem MMT, 1580
    zero matrix
        theorem MMZM, 1581
matrix product
    as composition of linear transformations
        example MPMR, 1582
matrix representation
    basis of eigenvectors
        example MRBE, 1583
    composition of linear transformations
        theorem MRCLT, 1584
    definition MR, 1585
    invertible
        theorem IMR, 1586
    multiple of a linear transformation
        theorem MRMLT, 1587
    notation, 1588
    restriction to generalized eigenspace
        theorem MRRGE, 1589
    sum of linear transformations
        theorem MRSLT, 1590
    theorem FTMR, 1591
    upper triangular
        theorem UTMR, 1592
matrix representations
    converting with change-of-basis
        example MRCM, 1593
    example OLTTR, 1594
matrix scalar multiplication
    example MSM, 1595
matrix vector space
    dimension
        theorem DM, 1596
matrix-adjoint product
    eigenvalues, eigenvectors
        theorem EEMAP, 1597
matrix-vector product
    example MTV, 1598
    notation, 1599
MBC (example), 1600
MBLT (theorem), 1601
MC (notation), 1602
MCC (subsection, section MO), 1603
MCCN (Property), 1604
MCF (Property), 1605
MCN (definition), 1606
MCN (subsection, section CNO), 1607
MCSM (example), 1608
MCT (theorem), 1609
MD (chapter), 1610
ME (definition), 1611
ME (notation), 1612
ME (subsection, section PEE), 1613
ME (technique, section PT), 1614
ME (theorem), 1615
ME.MMA (computation, section MMA), 1616
ME.SAGE (computation, section SAGE), 1617
ME.TI83 (computation, section TI83), 1618
ME.TI86 (computation, section TI86), 1619
MEASM (subsection, section MO), 1620
MFLT (example), 1621
MI (definition), 1622
MI (example), 1623
MI (notation), 1624
MI.MMA (computation, section MMA), 1625
MI.SAGE (computation, section SAGE), 1626
MICN (Property), 1627
MIF (Property), 1628
MIMI (theorem), 1629
MINM (section), 1630
MISLE (section), 1631
MISM (theorem), 1632
MIT (theorem), 1633
MIU (theorem), 1634
MIVS (example), 1635
MLT (subsection, section LT), 1636
MLTCV (theorem), 1637
MLTLT (theorem), 1638
MM (definition), 1639
MM (section), 1640
MM (subsection, section MM), 1641
MM.MMA (computation, section MMA), 1642
MMA (section), 1643
MMA (theorem), 1644
MMAD (theorem), 1645
MMCC (theorem), 1646
MMDAA (theorem), 1647
MMEE (subsection, section MM), 1648
MMIM (theorem), 1649
MMIP (theorem), 1650
MMNC (example), 1651
MMSMM (theorem), 1652
MMT (theorem), 1653
MMZM (theorem), 1654
MNEM (theorem), 1655
MNSLE (example), 1656
MO (section), 1657
MOLT (example), 1658
more variables than equations
    example OSGMD, 1659
    theorem CMVEI, 1660
MPMR (example), 1661
MR (definition), 1662
MR (notation), 1663
MR (section), 1664
MRBE (example), 1665
MRCB (theorem), 1666
MRCLT (diagram), 1667
MRCLT (theorem), 1668
MRCM (example), 1669
MRLS (definition), 1670
MRLS (notation), 1671
MRMLT (theorem), 1672
MRRGE (theorem), 1673
MRS (subsection, section CB), 1674
MRSLT (theorem), 1675
MSCN (example), 1676
MSM (definition), 1677
MSM (example), 1678
MSM (notation), 1679
MTV (example), 1680
multiplicative associativity
    complex numbers
        Property MACN, 1681
multiplicative closure
    complex numbers
        Property MCCN, 1682
    field
        Property MCF, 1683
multiplicative commuativity
    complex numbers
        Property CMCN, 1684
multiplicative inverse
    complex numbers
        Property MICN, 1685
MVNSE (subsection, section RREF), 1686
MVP (definition), 1687
MVP (notation), 1688
MVP (subsection, section MM), 1689
MVSLD (theorem), 1690
MWIAA (example), 1691

N (archetype), 1692
N (subsection, section O), 1693
N (technique, section PT), 1694
NDMS4 (example), 1695
negation of statements
    technique N, 1696
NEM (theorem), 1697
NI (theorem), 1698
NIAO (example), 1699
NIAQ (example), 1700
NIAQR (example), 1701
NIDAU (example), 1702
nilpotent
    linear transformation
        definition NLT, 1703
NILT (diagram), 1704
NJB (theorem), 1705
NJB5 (example), 1706
NKAO (example), 1707
NLT (definition), 1708
NLT (example), 1709
NLT (section), 1710
NLT (subsection, section NLT), 1711
NLTFO (subsection, section LT), 1712
NM (definition), 1713
NM (example), 1714
NM (section), 1715
NM (subsection, section NM), 1716
NM (subsection, section OD), 1717
NM62 (example), 1718
NM64 (example), 1719
NM83 (example), 1720
NME1 (theorem), 1721
NME2 (theorem), 1722
NME3 (theorem), 1723
NME4 (theorem), 1724
NME5 (theorem), 1725
NME6 (theorem), 1726
NME7 (theorem), 1727
NME8 (theorem), 1728
NME9 (theorem), 1729
NMI (subsection, section MINM), 1730
NMLIC (theorem), 1731
NMPEM (theorem), 1732
NMRRI (theorem), 1733
NMTNS (theorem), 1734
NMUS (theorem), 1735
NOILT (theorem), 1736
NOLT (definition), 1737
NOLT (notation), 1738
NOM (definition), 1739
NOM (notation), 1740
nonsingular
    columns as basis
        theorem CNMB, 1741
nonsingular matrices
    linearly independent columns
        theorem NMLIC, 1742
nonsingular matrix
    Archetype B
        example NM, 1743
    column space, 1744
    elemntary matrices
        theorem NMPEM, 1745
    equivalences
        theorem NME1, 1746
        theorem NME2, 1747
        theorem NME3, 1748
        theorem NME4, 1749
        theorem NME5, 1750
        theorem NME6, 1751
        theorem NME7, 1752
        theorem NME8, 1753
        theorem NME9, 1754
    matrix inverse, 1755
    null space
        example NSNM, 1756
    nullity, 1757
    product of nonsingular matrices
        theorem NPNT, 1758
    rank
        theorem RNNM, 1759
    row-reduced
        theorem NMRRI, 1760
    trivial null space
        theorem NMTNS, 1761
    unique solutions
        theorem NMUS, 1762
nonsingular matrix, row-reduced
    example NSR, 1763
norm
    example CNSV, 1764
    inner product, 1765
    notation, 1766
normal matrix
    definition NRML, 1767
    example ANM, 1768
    orthonormal basis, 1769
notation
    A, 1770
    AM, 1771
    AME, 1772
    C, 1773
    CCCV, 1774
    CCM, 1775
    CCN, 1776
    CNA, 1777
    CNE, 1778
    CNM, 1779
    CSM, 1780
    CV, 1781
    CVA, 1782
    CVC, 1783
    CVE, 1784
    CVSM, 1785
    D, 1786
    DM, 1787
    DS, 1788
    ELEM, 1789
    ES, 1790
    GES, 1791
    GME, 1792
    HI, 1793
    HID, 1794
    HP, 1795
    IE, 1796
    IM, 1797
    IP, 1798
    JB, 1799
    KLT, 1800
    LNS, 1801
    LT, 1802
    LTR, 1803
    M, 1804
    MA, 1805
    MC, 1806
    ME, 1807
    MI, 1808
    MR, 1809
    MRLS, 1810
    MSM, 1811
    MVP, 1812
    NOLT, 1813
    NOM, 1814
    NSM, 1815
    NV, 1816
    RLT, 1817
    RO, 1818
    ROLT, 1819
    ROM, 1820
    RREFA, 1821
    RSM, 1822
    SC, 1823
    SE, 1824
    SETM, 1825
    SI, 1826
    SM, 1827
    SRM, 1828
    SSET, 1829
    SSV, 1830
    SU, 1831
    SUV, 1832
    T, 1833
    TM, 1834
    VR, 1835
    VSCV, 1836
    VSM, 1837
    ZCV, 1838
    ZM, 1839
notation for a linear system
    example NSE, 1840
NPNT (theorem), 1841
NRFO (subsection, section MR), 1842
NRML (definition), 1843
NRREF (example), 1844
NS.MMA (computation, section MMA), 1845
NSAO (example), 1846
NSAQ (example), 1847
NSAQR (example), 1848
NSC2A (example), 1849
NSC2S (example), 1850
NSC2Z (example), 1851
NSDAT (example), 1852
NSDS (example), 1853
NSE (example), 1854
NSEAI (example), 1855
NSLE (example), 1856
NSLIL (example), 1857
NSM (definition), 1858
NSM (notation), 1859
NSM (subsection, section HSE), 1860
NSMS (theorem), 1861
NSNM (example), 1862
NSNM (subsection, section NM), 1863
NSR (example), 1864
NSS (example), 1865
NSSLI (subsection, section LI), 1866
Null space
    as a span
        example NSDS, 1867
null space
    Archetype I
        example NSEAI, 1868
    basis
        theorem BNS, 1869
    computation
        example CNS1, 1870
        example CNS2, 1871
    isomorphic to kernel, 1872
    linearly independent basis
        example LINSB, 1873
    mathematica, 1874
    matrix
        definition NSM, 1875
    nonsingular matrix, 1876
    notation, 1877
    singular matrix, 1878
    spanning set
        example SSNS, 1879
        theorem SSNS, 1880
    subspace
        theorem NSMS, 1881
null space span, linearly independent
    Archetype L
        example NSLIL, 1882
nullity
    computing, 1883
    injective linear transformation
        theorem NOILT, 1884
    linear transformation
        definition NOLT, 1885
    matrix, 1886
        definition NOM, 1887
    notation, 1888, 1889
    square matrix, 1890
NV (definition), 1891
NV (notation), 1892
NVM (theorem), 1893

O (archetype), 1894
O (Property), 1895
O (section), 1896
OBC (subsection, section B), 1897
OBNM (theorem), 1898
OBUTR (theorem), 1899
OC (Property), 1900
OCN (Property), 1901
OD (section), 1902
OD (subsection, section OD), 1903
OD (theorem), 1904
OF (Property), 1905
OLTTR (example), 1906
OM (Property), 1907
one
    column vectors
        Property OC, 1908
    complex numbers
        Property OCN, 1909
    field
        Property OF, 1910
    matrices
        Property OM, 1911
    vectors
        Property O, 1912
ONFV (example), 1913
ONS (definition), 1914
ONTV (example), 1915
orthogonal
    linear independence
        theorem OSLI, 1916
    set
        example AOS, 1917
    set of vectors
        definition OSV, 1918
    vector pairs
        definition OV, 1919
orthogonal vectors
    example TOV, 1920
orthonormal
    definition ONS, 1921
    matrix columns
        example OSMC, 1922
orthonormal basis
    normal matrix
        theorem OBNM, 1923
orthonormal diagonalization
    theorem OD, 1924
orthonormal set
    four vectors
        example ONFV, 1925
    three vectors
        example ONTV, 1926
OSGMD (example), 1927
OSIS (theorem), 1928
OSLI (theorem), 1929
OSMC (example), 1930
OSV (definition), 1931
OV (definition), 1932
OV (subsection, section O), 1933

P (appendix), 1934
P (archetype), 1935
P (technique, section PT), 1936
particular solutions
    example PSHS, 1937
PCNA (theorem), 1938
PCVS (example), 1939
PD (section), 1940
PDM (section), 1941
PDM (theorem), 1942
PEE (section), 1943
PEEF (theorem), 1944
PI (definition), 1945
PI (subsection, section LT), 1946
PI (technique, section PT), 1947
PIP (theorem), 1948
PM (example), 1949
PM (subsection, section EE), 1950
PMI (subsection, section MISLE), 1951
PMM (subsection, section MM), 1952
PMR (subsection, section MR), 1953
PNLT (subsection, section NLT), 1954
POD (section), 1955
polar decomposition
    theorem PDM, 1956
polynomial
    of a matrix
        example PM, 1957
polynomial vector space
    dimension
        theorem DP, 1958
positive semi-definite
    creating
        theorem CPSM, 1959
positive semi-definite matrix
    definition PSM, 1960
    eigenvalues
        theorem EPSM, 1961
practice
    technique P, 1962
pre-image
    definition PI, 1963
    kernel
        theorem KPI, 1964
pre-images
    example SPIAS, 1965
principal axis theorem, 1966
product of triangular matrices
    theorem PTMT, 1967
Property
    AA, 1968
    AAC, 1969
    AACN, 1970
    AAF, 1971
    AAM, 1972
    AC, 1973
    ACC, 1974
    ACCN, 1975
    ACF, 1976
    ACM, 1977
    AI, 1978
    AIC, 1979
    AICN, 1980
    AIF, 1981
    AIM, 1982
    C, 1983
    CACN, 1984
    CAF, 1985
    CC, 1986
    CM, 1987
    CMCN, 1988
    CMF, 1989
    DCN, 1990
    DF, 1991
    DMAM, 1992
    DSA, 1993
    DSAC, 1994
    DSAM, 1995
    DVA, 1996
    DVAC, 1997
    MACN, 1998
    MAF, 1999
    MCCN, 2000
    MCF, 2001
    MICN, 2002
    MIF, 2003
    O, 2004
    OC, 2005
    OCN, 2006
    OF, 2007
    OM, 2008
    SC, 2009
    SCC, 2010
    SCM, 2011
    SMA, 2012
    SMAC, 2013
    SMAM, 2014
    Z, 2015
    ZC, 2016
    ZCN, 2017
    ZF, 2018
    ZM, 2019
PSHS (example), 2020
PSHS (subsection, section LC), 2021
PSM (definition), 2022
PSM (section), 2023
PSM (subsection, section PSM), 2024
PSM (subsection, section SD), 2025
PSMSR (theorem), 2026
PSPHS (theorem), 2027
PSS (subsection, section SSLE), 2028
PSSD (theorem), 2029
PSSLS (theorem), 2030
PT (section), 2031
PTFP (example), 2032
PTM (example), 2033
PTMEE (example), 2034
PTMT (theorem), 2035

Q (archetype), 2036

R (acronyms, section JCF), 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
    ti83, 2153
    ti86, 2154
row space
    Archetype I
        example RSAI, 2155
    as column space, 2156
    basis
        example RSB, 2157
        theorem BRS, 2158
    matrix, 2159
    notation, 2160
    row-equivalent matrices
        theorem REMRS, 2161
    subspace
        theorem RSMS, 2162
row-equivalent matrices
    definition REM, 2163
    example TREM, 2164
    row space, 2165
    row spaces
        example RSREM, 2166
    theorem REMES, 2167
row-reduce
    the verb
        definition RR, 2168
row-reduced matrices
    theorem REMEF, 2169
RPI (theorem), 2170
RPNC (theorem), 2171
RPNDD (theorem), 2172
RR (definition), 2173
RR.MMA (computation, section MMA), 2174
RR.TI83 (computation, section TI83), 2175
RR.TI86 (computation, section TI86), 2176
RREF (definition), 2177
RREF (example), 2178
RREF (section), 2179
RREF (subsection, section RREF), 2180
RREFA (notation), 2181
RREFN (example), 2182
RREFU (theorem), 2183
RRTI (example), 2184
RS (example), 2185
RSAI (example), 2186
RSB (example), 2187
RSC5 (example), 2188
RSLT (theorem), 2189
RSM (definition), 2190
RSM (notation), 2191
RSM (subsection, section CRS), 2192
RSMS (theorem), 2193
RSNS (example), 2194
RSREM (example), 2195
RSSC4 (example), 2196
RT (subsection, section PD), 2197
RVMR (example), 2198

S (archetype), 2199
S (definition), 2200
S (example), 2201
S (section), 2202
SAA (example), 2203
SAB (example), 2204
SABMI (example), 2205
SAE (example), 2206
sage
    eigenspaces (computation), 2207
    matrix entry (computation), 2208
    matrix inverse (computation), 2209
    rings (computation), 2210
    transpose of a matrix (computation), 2211
SAGE (section), 2212
SAN (example), 2213
SAR (example), 2214
SAS (section), 2215
SAV (example), 2216
SC (definition), 2217
SC (example), 2218
SC (notation), 2219
SC (Property), 2220
SC (subsection, section S), 2221
SC (subsection, section SET), 2222
SC3 (example), 2223
SCAA (example), 2224
SCAB (example), 2225
SCAD (example), 2226
scalar closure
    column vectors
        Property SCC, 2227
    matrices
        Property SCM, 2228
    vectors
        Property SC, 2229
scalar multiple
    matrix inverse, 2230
scalar multiplication
    zero scalar
        theorem ZSSM, 2231
    zero vector
        theorem ZVSM, 2232
    zero vector result
        theorem SMEZV, 2233
scalar multiplication associativity
    column vectors
        Property SMAC, 2234
    matrices
        Property SMAM, 2235
    vectors
        Property SMA, 2236
SCB (theorem), 2237
SCC (Property), 2238
SCM (Property), 2239
SD (section), 2240
SDS (example), 2241
SE (definition), 2242
SE (notation), 2243
secret sharing
    6 ways
        example SS6W, 2244
SEE (example), 2245
SEEF (example), 2246
SER (theorem), 2247
set
    cardinality
        definition C, 2248
        example CS, 2249
        notation, 2250
    complement
        definition SC, 2251
        example SC, 2252
        notation, 2253
    definition SET, 2254
    empty
        definition ES, 2255
    equality
        definition SE, 2256
        notation, 2257
    intersection
        definition SI, 2258
        example SI, 2259
        notation, 2260
    membership
        example SETM, 2261
        notation, 2262
    size, 2263
    subset, 2264
    union
        definition SU, 2265
        example SU, 2266
        notation, 2267
SET (definition), 2268
SET (section), 2269
SETM (example), 2270
SETM (notation), 2271
shoes, 2272
SHS (subsection, section HSE), 2273
SI (definition), 2274
SI (example), 2275
SI (notation), 2276
SI (subsection, section IVLT), 2277
SIM (definition), 2278
similar matrices
    equal eigenvalues
        example EENS, 2279
    eual eigenvalues
        theorem SMEE, 2280
    example SMS3, 2281
    example SMS5, 2282
similarity
    definition SIM, 2283
    equivalence relation
        theorem SER, 2284
singular matrix
    Archetype A
        example S, 2285
    null space
        example NSS, 2286
singular matrix, row-reduced
    example SRR, 2287
singular value decomposition
    theorem SVD, 2288
singular values
    definition SV, 2289
SLE (acronyms, section NM), 2290
SLE (chapter), 2291
SLE (definition), 2292
SLE (subsection, section SSLE), 2293
SLELT (subsection, section IVLT), 2294
SLEMM (theorem), 2295
SLSLC (theorem), 2296
SLT (definition), 2297
SLT (section), 2298
SLTB (theorem), 2299
SLTD (subsection, section SLT), 2300
SLTD (theorem), 2301
SLTLT (theorem), 2302
SM (definition), 2303
SM (notation), 2304
SM (subsection, section SD), 2305
SM2Z7 (example), 2306
SM32 (example), 2307
SMA (Property), 2308
SMAC (Property), 2309
SMAM (Property), 2310
SMEE (theorem), 2311
SMEZV (theorem), 2312
SMLT (example), 2313
SMS (theorem), 2314
SMS3 (example), 2315
SMS5 (example), 2316
SMZD (theorem), 2317
SMZE (theorem), 2318
SNCM (theorem), 2319
SO (subsection, section SET), 2320
socks, 2321
SOL (subsection, section B), 2322
SOL (subsection, section CB), 2323
SOL (subsection, section CRS), 2324
SOL (subsection, section D), 2325
SOL (subsection, section DM), 2326
SOL (subsection, section EE), 2327
SOL (subsection, section F), 2328
SOL (subsection, section FS), 2329
SOL (subsection, section HSE), 2330
SOL (subsection, section ILT), 2331
SOL (subsection, section IVLT), 2332
SOL (subsection, section LC), 2333
SOL (subsection, section LDS), 2334
SOL (subsection, section LI), 2335
SOL (subsection, section LISS), 2336
SOL (subsection, section LT), 2337
SOL (subsection, section MINM), 2338
SOL (subsection, section MISLE), 2339
SOL (subsection, section MM), 2340
SOL (subsection, section MO), 2341
SOL (subsection, section MR), 2342
SOL (subsection, section NM), 2343
SOL (subsection, section O), 2344
SOL (subsection, section PD), 2345
SOL (subsection, section PDM), 2346
SOL (subsection, section PEE), 2347
SOL (subsection, section RREF), 2348
SOL (subsection, section S), 2349
SOL (subsection, section SD), 2350
SOL (subsection, section SLT), 2351
SOL (subsection, section SS), 2352
SOL (subsection, section SSLE), 2353
SOL (subsection, section T), 2354
SOL (subsection, section TSS), 2355
SOL (subsection, section VO), 2356
SOL (subsection, section VR), 2357
SOL (subsection, section VS), 2358
SOL (subsection, section WILA), 2359
solution set
    Archetype A
        example SAA, 2360
    archetype E
        example SAE, 2361
    theorem PSPHS, 2362
solution sets
    possibilities
        theorem PSSLS, 2363
solution vector
    definition SOLV, 2364
SOLV (definition), 2365
solving homogeneous system
    Archetype A
        example HISAA, 2366
    Archetype B
        example HUSAB, 2367
    Archetype D
        example HISAD, 2368
solving nonlinear equations
    example STNE, 2369
SP4 (example), 2370
span
    basic
        example ABS, 2371
    basis
        theorem BS, 2372
    definition SS, 2373
    definition SSCV, 2374
    improved
        example IAS, 2375
    notation, 2376
    reducing
        example RSSC4, 2377
    reduction
        example RS, 2378
    removing vectors
        example COV, 2379
    reworking elements
        example RES, 2380
    set of polynomials
        example SSP, 2381
    subspace
        theorem SSS, 2382
span of columns
    Archetype A
        example SCAA, 2383
    Archetype B
        example SCAB, 2384
    Archetype D
        example SCAD, 2385
spanning set
    crazy vector space
        example SSC, 2386
    definition TSVS, 2387
    matrices
        example SSM22, 2388
    more vectors
        theorem SSLD, 2389
    polynomials
        example SSP4, 2390
SPIAS (example), 2391
SQM (definition), 2392
square root
    eigenvalues, eigenspaces
        theorem EESR, 2393
    matrix
        definition SRM, 2394
        notation, 2395
    positive semi-definite matrix
        theorem PSMSR, 2396
    unique
        theorem USR, 2397
SR (section), 2398
SRM (definition), 2399
SRM (notation), 2400
SRM (subsection, section SR), 2401
SRR (example), 2402
SS (definition), 2403
SS (example), 2404
SS (section), 2405
SS (subsection, section LISS), 2406
SS (theorem), 2407
SS6W (example), 2408
SSC (example), 2409
SSCV (definition), 2410
SSET (definition), 2411
SSET (example), 2412
SSET (notation), 2413
SSLD (theorem), 2414
SSLE (section), 2415
SSM22 (example), 2416
SSNS (example), 2417
SSNS (subsection, section SS), 2418
SSNS (theorem), 2419
SSP (example), 2420
SSP4 (example), 2421
SSRLT (theorem), 2422
SSS (theorem), 2423
SSSLT (subsection, section SLT), 2424
SSV (notation), 2425
SSV (subsection, section SS), 2426
standard unit vector
    notation, 2427
starting proofs
    technique GS, 2428
STLT (example), 2429
STNE (example), 2430
SU (definition), 2431
SU (example), 2432
SU (notation), 2433
submatrix
    notation, 2434
subset
    definition SSET, 2435
    notation, 2436
subspace
    as null space
        example RSNS, 2437
    characterized
        example ASC, 2438
    definition S, 2439
    in P4
        example SP4, 2440
    not, additive closure
        example NSC2A, 2441
    not, scalar closure
        example NSC2S, 2442
    not, zero vector
        example NSC2Z, 2443
    testing
        theorem TSS, 2444
    trivial
        definition TS, 2445
    verification
        example SC3, 2446
        example SM32, 2447
subspaces
    equal dimension
        theorem EDYES, 2448
surjective
    Archetype N
        example SAN, 2449
    example SAR, 2450
    not
        example NSAQ, 2451
        example NSAQR, 2452
    not, Archetype O
        example NSAO, 2453
    not, by dimension
        example NSDAT, 2454
    polynomials to matrices
        example SAV, 2455
surjective linear transformation
    bases
        theorem SLTB, 2456
surjective linear transformations
    dimension
        theorem SLTD, 2457
SUV (definition), 2458
SUV (notation), 2459
SUVB (theorem), 2460
SUVOS (example), 2461
SV (definition), 2462
SVD (section), 2463
SVD (subsection, section SVD), 2464
SVD (theorem), 2465
SVP4 (example), 2466
SYM (definition), 2467
SYM (example), 2468
symmetric matrices
    theorem SMS, 2469
symmetric matrix
    example SYM, 2470
system of equations
    vector equality
        example VESE, 2471
system of linear equations
    definition SLE, 2472

T (archetype), 2473
T (definition), 2474
T (notation), 2475
T (part), 2476
T (section), 2477
T (technique, section PT), 2478
TCSD (example), 2479
TD (section), 2480
TD (subsection, section TD), 2481
TD (theorem), 2482
TD4 (example), 2483
TDEE (theorem), 2484
TDEE6 (example), 2485
TDSSE (example), 2486
TDSSE (subsection, section TD), 2487
technique
    C, 2488
    CD, 2489
    CP, 2490
    CV, 2491
    D, 2492
    DC, 2493
    E, 2494
    GS, 2495
    I, 2496
    L, 2497
    LC, 2498
    ME, 2499
    N, 2500
    P, 2501
    PI, 2502
    T, 2503
    U, 2504
theorem
    AA, 2505
    AIP, 2506
    AISM, 2507
    AIU, 2508
    AMA, 2509
    AMSM, 2510
    BCS, 2511
    BIS, 2512
    BNS, 2513
    BRS, 2514
    BS, 2515
    CB, 2516
    CCM, 2517
    CCRA, 2518
    CCRM, 2519
    CCT, 2520
    CFDVS, 2521
    CFNLT, 2522
    CHT, 2523
    CILTI, 2524
    CINM, 2525
    CIVLT, 2526
    CLI, 2527
    CLTLT, 2528
    CMVEI, 2529
    CNMB, 2530
    COB, 2531
    CPSM, 2532
    CRMA, 2533
    CRMSM, 2534
    CRN, 2535
    CRSM, 2536
    CRVA, 2537
    CSCS, 2538
    CSLTS, 2539
    CSMS, 2540
    CSNM, 2541
    CSRN, 2542
    CSRST, 2543
    CSS, 2544
    CUMOS, 2545
    DC, 2546
    DCM, 2547
    DCP, 2548
    DEC, 2549
    DED, 2550
    DEM, 2551
    DEMMM, 2552
    DER, 2553
    DERC, 2554
    DFS, 2555
    DGES, 2556
    DIM, 2557
    DLDS, 2558
    DM, 2559
    DMFE, 2560
    DMHP, 2561
    DMMP, 2562
    DMST, 2563
    DNLT, 2564
    DP, 2565
    DRCM, 2566
    DRCMA, 2567
    DRCS, 2568
    DRMM, 2569
    DSD, 2570
    DSFB, 2571
    DSFOS, 2572
    DSLI, 2573
    DSZI, 2574
    DSZV, 2575
    DT, 2576
    DVM, 2577
    DZRC, 2578
    EDELI, 2579
    EDYES, 2580
    EEMAP, 2581
    EER, 2582
    EESR, 2583
    EIM, 2584
    EIS, 2585
    ELIS, 2586
    EMDRO, 2587
    EMHE, 2588
    EMMVP, 2589
    EMN, 2590
    EMNS, 2591
    EMP, 2592
    EMRCP, 2593
    EMS, 2594
    ENLT, 2595
    EOMP, 2596
    EOPSS, 2597
    EPM, 2598
    EPSM, 2599
    ERMCP, 2600
    ESMM, 2601
    ETM, 2602
    FIMP, 2603
    FS, 2604
    FTMR, 2605
    FVCS, 2606
    G, 2607
    GEK, 2608
    GESD, 2609
    GESIS, 2610
    GSP, 2611
    HMIP, 2612
    HMOE, 2613
    HMRE, 2614
    HMVEI, 2615
    HPC, 2616
    HPDAA, 2617
    HPHI, 2618
    HPHID, 2619
    HPSMM, 2620
    HSC, 2621
    ICBM, 2622
    ICLT, 2623
    IFDVS, 2624
    IILT, 2625
    ILTB, 2626
    ILTD, 2627
    ILTIS, 2628
    ILTLI, 2629
    ILTLT, 2630
    IMILT, 2631
    IMR, 2632
    IP, 2633
    IPAC, 2634
    IPN, 2635
    IPSM, 2636
    IPVA, 2637
    ISRN, 2638
    ITMT, 2639
    IVSED, 2640
    JCFLT, 2641
    KILT, 2642
    KLTS, 2643
    KNSI, 2644
    KPI, 2645
    KPIS, 2646
    KPLT, 2647
    KPNLT, 2648
    LIVHS, 2649
    LIVRN, 2650
    LNSMS, 2651
    LSMR, 2652
    LTDB, 2653
    LTLC, 2654
    LTTZZ, 2655
    MBLT, 2656
    MCT, 2657
    ME, 2658
    MIMI, 2659
    MISM, 2660
    MIT, 2661
    MIU, 2662
    MLTCV, 2663
    MLTLT, 2664
    MMA, 2665
    MMAD, 2666
    MMCC, 2667
    MMDAA, 2668
    MMIM, 2669
    MMIP, 2670
    MMSMM, 2671
    MMT, 2672
    MMZM, 2673
    MNEM, 2674
    MRCB, 2675
    MRCLT, 2676
    MRMLT, 2677
    MRRGE, 2678
    MRSLT, 2679
    MVSLD, 2680
    NEM, 2681
    NI, 2682
    NJB, 2683
    NME1, 2684
    NME2, 2685
    NME3, 2686
    NME4, 2687
    NME5, 2688
    NME6, 2689
    NME7, 2690
    NME8, 2691
    NME9, 2692
    NMLIC, 2693
    NMPEM, 2694
    NMRRI, 2695
    NMTNS, 2696
    NMUS, 2697
    NOILT, 2698
    NPNT, 2699
    NSMS, 2700
    NVM, 2701
    OBNM, 2702
    OBUTR, 2703
    OD, 2704
    OSIS, 2705
    OSLI, 2706
    PCNA, 2707
    PDM, 2708
    PEEF, 2709
    PIP, 2710
    PSMSR, 2711
    PSPHS, 2712
    PSSD, 2713
    PSSLS, 2714
    PTMT, 2715
    RCLS, 2716
    RCSI, 2717
    RDS, 2718
    REMEF, 2719
    REMES, 2720
    REMRS, 2721
    RGEN, 2722
    RLTS, 2723
    RMRT, 2724
    RNNM, 2725
    ROD, 2726
    ROSLT, 2727
    RPI, 2728
    RPNC, 2729
    RPNDD, 2730
    RREFU, 2731
    RSLT, 2732
    RSMS, 2733
    SCB, 2734
    SER, 2735
    SLEMM, 2736
    SLSLC, 2737
    SLTB, 2738
    SLTD, 2739
    SLTLT, 2740
    SMEE, 2741
    SMEZV, 2742
    SMS, 2743
    SMZD, 2744
    SMZE, 2745
    SNCM, 2746
    SS, 2747
    SSLD, 2748
    SSNS, 2749
    SSRLT, 2750
    SSS, 2751
    SUVB, 2752
    SVD, 2753
    TD, 2754
    TDEE, 2755
    technique T, 2756
    TIST, 2757
    TL, 2758
    TMA, 2759
    TMSM, 2760
    TSE, 2761
    TSRM, 2762
    TSS, 2763
    TT, 2764
    TTMI, 2765
    UMCOB, 2766
    UMI, 2767
    UMPIP, 2768
    USR, 2769
    UTMR, 2770
    VFSLS, 2771
    VRI, 2772
    VRILT, 2773
    VRLT, 2774
    VRRB, 2775
    VRS, 2776
    VSLT, 2777
    VSPCV, 2778
    VSPM, 2779
    ZSSM, 2780
    ZVSM, 2781
    ZVU, 2782
ti83
    matrix entry (computation), 2783
    row reduce (computation), 2784
    vector linear combinations (computation), 2785
TI83 (section), 2786
ti86
    matrix entry (computation), 2787
    row reduce (computation), 2788
    transpose of a matrix (computation), 2789
    vector linear combinations (computation), 2790
TI86 (section), 2791
TIS (example), 2792
TIST (theorem), 2793
TIVS (example), 2794
TKAP (example), 2795
TL (theorem), 2796
TLC (example), 2797
TM (definition), 2798
TM (example), 2799
TM (notation), 2800
TM (subsection, section OD), 2801
TM.MMA (computation, section MMA), 2802
TM.SAGE (computation, section SAGE), 2803
TM.TI86 (computation, section TI86), 2804
TMA (theorem), 2805
TMP (example), 2806
TMSM (theorem), 2807
TOV (example), 2808
trace
    definition T, 2809
    linearity
        theorem TL, 2810
    matrix multiplication
        theorem TSRM, 2811
    notation, 2812
    similarity
        theorem TIST, 2813
    sum of eigenvalues
        theorem TSE, 2814
trail mix
    example TMP, 2815
transpose
    matrix scalar multiplication
        theorem TMSM, 2816
    example TM, 2817
    matrix addition
        theorem TMA, 2818
    matrix inverse, 2819, 2820
    notation, 2821
    scalar multiplication, 2822
transpose of a matrix
    mathematica, 2823
    sage, 2824
    ti86, 2825
transpose of a transpose
    theorem TT, 2826
TREM (example), 2827
triangular decomposition
    entry by entry, size 6
        example TDEE6, 2828
    entry by entry
        theorem TDEE, 2829
    size 4
        example TD4, 2830
    solving systems of equations
        example TDSSE, 2831
    theorem TD, 2832
triangular matrix
    inverse
        theorem ITMT, 2833
trivial solution
    system of equations
        definition TSHSE, 2834
TS (definition), 2835
TS (subsection, section S), 2836
TSE (theorem), 2837
TSHSE (definition), 2838
TSM (subsection, section MO), 2839
TSRM (theorem), 2840
TSS (section), 2841
TSS (subsection, section S), 2842
TSS (theorem), 2843
TSVS (definition), 2844
TT (theorem), 2845
TTMI (theorem), 2846
TTS (example), 2847
typical systems, 2 × 2
    example TTS, 2848

U (archetype), 2849
U (technique, section PT), 2850
UM (definition), 2851
UM (subsection, section MINM), 2852
UM3 (example), 2853
UMCOB (theorem), 2854
UMI (theorem), 2855
UMPIP (theorem), 2856
unique solution, 3 × 3
    example US, 2857
    example USR, 2858
uniqueness
    technique U, 2859
unit vectors
    basis
        theorem SUVB, 2860
    definition SUV, 2861
    orthogonal
        example SUVOS, 2862
unitary
    permutation matrix
        example UPM, 2863
    size 3
        example UM3, 2864
unitary matrices
    columns
        theorem CUMOS, 2865
unitary matrix
    inner product
        theorem UMPIP, 2866
UPM (example), 2867
upper triangular matrix
    definition UTM, 2868
US (example), 2869
USR (example), 2870
USR (theorem), 2871
UTM (definition), 2872
UTMR (subsection, section OD), 2873
UTMR (theorem), 2874

V (acronyms, section O), 2875
V (archetype), 2876
V (chapter), 2877
VA (example), 2878
Vandermonde matrix
    definition VM, 2879
vandermonde matrix
    determinant
        theorem DVM, 2880
    nonsingular
        theorem NVM, 2881
    size 4
        example VM4, 2882
VEASM (subsection, section VO), 2883
vector
    addition
        definition CVA, 2884
    column
        definition CV, 2885
    equality
        definition CVE, 2886
        notation, 2887
    inner product
        definition IP, 2888
    norm
        definition NV, 2889
    notation, 2890
    of constants
        definition VOC, 2891
    product with matrix, 2892, 2893
    scalar multiplication
        definition CVSM, 2894
vector addition
    example VA, 2895
vector component
    notation, 2896
vector form of solutions
    Archetype D
        example VFSAD, 2897
    Archetype I
        example VFSAI, 2898
    Archetype L
        example VFSAL, 2899
    example VFS, 2900
    mathematica, 2901
    theorem VFSLS, 2902
vector linear combinations
    mathematica, 2903
    ti83, 2904
    ti86, 2905
vector representation
    example AVR, 2906
    example VRC4, 2907
    injective
        theorem VRI, 2908
    invertible
        theorem VRILT, 2909
    linear transformation
        definition VR, 2910
        notation, 2911
        theorem VRLT, 2912
    surjective
        theorem VRS, 2913
    theorem VRRB, 2914
vector representations
    polynomials
        example VRP2, 2915
vector scalar multiplication
    example CVSM, 2916
vector space
    characterization
        theorem CFDVS, 2917
    column vectors
        definition VSCV, 2918
    definition VS, 2919
    infinite dimension
        example VSPUD, 2920
    linear transformations
        theorem VSLT, 2921
    over integers mod 5
        example VSIM5, 2922
vector space of column vectors
    notation, 2923
vector space of functions
    example VSF, 2924
vector space of infinite sequences
    example VSIS, 2925
vector space of matrices
    definition VSM, 2926
    example VSM, 2927
    notation, 2928
vector space of polynomials
    example VSP, 2929
vector space properties
    column vectors
        theorem VSPCV, 2930
    matrices
        theorem VSPM, 2931
vector space, crazy
    example CVS, 2932
vector space, singleton
    example VSS, 2933
vector spaces
    isomorphic
        definition IVS, 2934
        theorem IFDVS, 2935
VESE (example), 2936
VFS (example), 2937
VFSAD (example), 2938
VFSAI (example), 2939
VFSAL (example), 2940
VFSLS (theorem), 2941
VFSS (subsection, section LC), 2942
VFSS.MMA (computation, section MMA), 2943
VLC.MMA (computation, section MMA), 2944
VLC.TI83 (computation, section TI83), 2945
VLC.TI86 (computation, section TI86), 2946
VM (definition), 2947
VM (section), 2948
VM4 (example), 2949
VO (section), 2950
VOC (definition), 2951
VR (definition), 2952
VR (notation), 2953
VR (section), 2954
VR (subsection, section LISS), 2955
VRC4 (example), 2956
VRI (theorem), 2957
VRILT (theorem), 2958
VRLT (theorem), 2959
VRP2 (example), 2960
VRRB (theorem), 2961
VRS (theorem), 2962
VS (acronyms, section PD), 2963
VS (chapter), 2964
VS (definition), 2965
VS (section), 2966
VS (subsection, section VS), 2967
VSCV (definition), 2968
VSCV (example), 2969
VSCV (notation), 2970
VSF (example), 2971
VSIM5 (example), 2972
VSIS (example), 2973
VSLT (theorem), 2974
VSM (definition), 2975
VSM (example), 2976
VSM (notation), 2977
VSP (example), 2978
VSP (subsection, section MO), 2979
VSP (subsection, section VO), 2980
VSP (subsection, section VS), 2981
VSPCV (theorem), 2982
VSPM (theorem), 2983
VSPUD (example), 2984
VSS (example), 2985

W (archetype), 2986
WILA (section), 2987

X (archetype), 2988

Z (Property), 2989
ZC (Property), 2990
ZCN (Property), 2991
ZCV (definition), 2992
ZCV (notation), 2993
zero
    complex numbers
        Property ZCN, 2994
    field
        Property ZF, 2995
zero column vector
    definition ZCV, 2996
    notation, 2997
zero matrix
    notation, 2998
zero vector
    column vectors
        Property ZC, 2999
    matrices
        Property ZM, 3000
    unique
        theorem ZVU, 3001
    vectors
        Property Z, 3002
ZF (Property), 3003
ZM (definition), 3004
ZM (notation), 3005
ZM (Property), 3006
ZNDAB (example), 3007
ZSSM (theorem), 3008
ZVSM (theorem), 3009
ZVU (theorem), 3010