## 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
column vectors
Property AAC, 22
complex numbers
Property AACN, 23
matrices
Property AAM, 24
vectors
Property AA, 25
column vectors
Property ACC, 26
complex numbers
Property ACCN, 27
field
Property ACF, 28
matrices
Property ACM, 29
vectors
Property AC, 30
complex numbers
Property CACN, 31
complex numbers
Property AICN, 32
from scalar multiplication
theorem AISM, 33
column vectors
Property AIC, 34
matrices
Property AIM, 35
unique
theorem AIU, 36
vectors
Property AI, 37
definition A, 38
inner product
theorem AIP, 39
notation, 40
of a matrix sum
theorem AMA, 41
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
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
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
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
theorem CRVA, 267
conjugate of a vector
notation, 268
conjugation
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
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
matrices
Property DMAM, 576
column vectors
Property DSAC, 577
matrices
Property DSAM, 578
vectors
Property DSA, 579
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
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
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
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
notation, 1145
definition HID, 1146
notation, 1147
definition HI, 1148
Diagonalizable Matrices
theorem DMHP, 1149
notation, 1150
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
HM (definition), 1167
HM (subsection, section MM), 1168
HMEM5 (example), 1169
HMIP (theorem), 1170
HMOE (theorem), 1171
HMRE (theorem), 1172
HMVEI (theorem), 1173
homogeneous system
Archetype C
example AHSAC, 1174
consistent
theorem HSC, 1175
definition HS, 1176
infinitely many solutions
theorem HMVEI, 1177
homogeneous systems
linear independence, 1178
HP (definition), 1179
HP (example), 1180
HP (notation), 1181
HP (section), 1182
HPC (theorem), 1183
HPDAA (theorem), 1184
HPDM (example), 1185
HPHI (theorem), 1186
HPHID (theorem), 1187
HPSMM (theorem), 1188
HS (definition), 1189
HSC (theorem), 1190
HSE (section), 1191
HUSAB (example), 1192

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

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

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

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

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

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

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

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

Q (archetype), 2038

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

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

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

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

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

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

X (archetype), 2997

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