B (archetype), 97
B (definition), 98
B (section), 99
B (subsection, section B), 100
basis
columns nonsingular matrix
example CABAK, 101
common size
theorem BIS, 102
crazy vector apace
example BC, 103
definition B, 104
matrices
example BM, 105
example BSM22, 106
polynomials
example BP, 107
example BPR, 108
example BSP4, 109
example SVP4, 110
subspace of matrices
example BDM22, 111
BC (example), 112
BCS (theorem), 113
BDE (example), 114
BDM22 (example), 115
best cities
money magazine
example MBC, 116
BIS (theorem), 117
BM (example), 118
BNM (subsection, section B), 119
BNS (theorem), 120
BP (example), 121
BPR (example), 122
BRLT (example), 123
BRS (theorem), 124
BS (theorem), 125
BSCV (subsection, section B), 126
BSM22 (example), 127
BSP4 (example), 128
C (archetype), 129
C (definition), 130
C (notation), 131
C (part), 132
C (Property), 133
C (technique, section PT), 134
CABAK (example), 135
CAEHW (example), 136
canonical form
nilpotent linear transformation
example CFNLT, 137
theorem CFNLT, 138
CAV (subsection, section O), 139
Cayley-Hamilton
theorem CHT, 140
CB (section), 141
CB (theorem), 142
CBCV (example), 143
CBM (definition), 144
CBM (subsection, section CB), 145
CBOB (theorem), 146
CBP (example), 147
CC (Property), 148
CCCV (definition), 149
CCCV (notation), 150
CCM (definition), 151
CCM (example), 152
CCM (notation), 153
CCM (theorem), 154
CCN (definition), 155
CCN (notation), 156
CCN (subsection, section CNO), 157
CCRA (theorem), 158
CCRM (theorem), 159
CCT (theorem), 160
CD (subsection, section DM), 161
CD (technique, section PT), 162
CEE (subsection, section EE), 163
CELT (example), 164
CELT (subsection, section CB), 165
CEMS6 (example), 166
CF (section), 167
CFDVS (theorem), 168
CFNLT (example), 169
CFNLT (subsection, section NLT), 170
CFNLT (theorem), 171
CFV (example), 172
change of basis
between polynomials
example CBP, 173
change-of-basis
between column vectors
example CBCV, 174
matrix representation
theorem MRCB, 175
similarity
theorem SCB, 176
theorem CB, 177
change-of-basis matrix
definition CBM, 178
inverse
theorem ICBM, 179
characteristic polynomial
definition CP, 180
degree
theorem DCP, 181
size 3 matrix
example CPMS3, 182
CHT (subsection, section JCF), 183
CHT (theorem), 184
CILT (subsection, section ILT), 185
CILTI (theorem), 186
CIM (subsection, section MISLE), 187
CINM (theorem), 188
CIVLT (theorem), 189
CLI (theorem), 190
CLTLT (theorem), 191
CM (definition), 192
CM (Property), 193
CM32 (example), 194
CMI (example), 195
CMIAB (example), 196
CMVEI (theorem), 197
CN (appendix), 198
CNA (definition), 199
CNA (notation), 200
CNA (subsection, section CNO), 201
CNE (definition), 202
CNE (notation), 203
CNM (definition), 204
CNM (notation), 205
CNMB (theorem), 206
CNO (section), 207
CNS1 (example), 208
CNS2 (example), 209
CNSV (example), 210
COB (theorem), 211
coefficient matrix
definition CM, 212
nonsingular
theorem SNCM, 213
column space
as null space
theorem FS, 214
Archetype A
example CSAA, 215
Archetype B
example CSAB, 216
as null space
example CSANS, 217
as null space, Archetype G
example FSAG, 218
as row space
theorem CSRST, 219
basis
theorem BCS, 220
consistent system
theorem CSCS, 221
consistent systems
example CSMCS, 222
isomorphic to range, 223
matrix, 224
nonsingular matrix
theorem CSNM, 225
notation, 226
original columns, Archetype D
example CSOCD, 227
row operations, Archetype I
example CSROI, 228
subspace
theorem CSMS, 229
testing membership
example MCSM, 230
two computations
example CSTW, 231
column vector addition
notation, 232
column vector scalar multiplication
notation, 233
commutativity
column vectors
Property CC, 234
matrices
Property CM, 235
vectors
Property C, 236
complex -space
example VSCV, 237
complex arithmetic
example ACN, 238
complex number
conjugate
example CSCN, 239
modulus
example MSCN, 240
complex number
conjugate
definition CCN, 241
modulus
definition MCN, 242
complex numbers
addition
definition CNA, 243
notation, 244
arithmetic properties
theorem PCNA, 245
equality
definition CNE, 246
notation, 247
multiplication
definition CNM, 248
notation, 249
complex vector space
dimension
theorem DCM, 250
composition
injective linear transformations
theorem CILTI, 251
surjective linear transformations
theorem CSLTS, 252
conjugate
addition
theorem CCRA, 253
column vector
definition CCCV, 254
matrix
definition CCM, 255
notation, 256
multiplication
theorem CCRM, 257
notation, 258
of conjugate of a matrix
theorem CCM, 259
scalar multiplication
theorem CRSM, 260
twice
theorem CCT, 261
vector addition
theorem CRVA, 262
conjugate of a vector
notation, 263
conjugation
matrix addition
theorem CRMA, 264
matrix scalar multiplication
theorem CRMSM, 265
matrix transpose
theorem MCT, 266
consistent linear system, 267
consistent linear systems
theorem CSRN, 268
consistent system
definition CS, 269
constructive proofs
technique C, 270
contradiction
technique CD, 271
contrapositive
technique CP, 272
converse
technique CV, 273
coordinates
orthonormal basis
theorem COB, 274
coordinatization
linear combination of matrices
example CM32, 275
linear independence
theorem CLI, 276
orthonormal basis
example CROB3, 277
example CROB4, 278
spanning sets
theorem CSS, 279
coordinatization principle, 280
coordinatizing
polynomials
example CP2, 281
COV (example), 282
COV (subsection, section LDS), 283
CP (definition), 284
CP (subsection, section VR), 285
CP (technique, section PT), 286
CP2 (example), 287
CPMS3 (example), 288
CPSM (theorem), 289
crazy vector space
example CVSR, 290
properties
example PCVS, 291
CRMA (theorem), 292
CRMSM (theorem), 293
CRN (theorem), 294
CROB3 (example), 295
CROB4 (example), 296
CRS (section), 297
CRS (subsection, section FS), 298
CRSM (theorem), 299
CRVA (theorem), 300
CS (definition), 301
CS (example), 302
CS (subsection, section TSS), 303
CSAA (example), 304
CSAB (example), 305
CSANS (example), 306
CSCN (example), 307
CSCS (theorem), 308
CSIP (example), 309
CSLT (subsection, section SLT), 310
CSLTS (theorem), 311
CSM (definition), 312
CSM (notation), 313
CSMCS (example), 314
CSMS (theorem), 315
CSNM (subsection, section CRS), 316
CSNM (theorem), 317
CSOCD (example), 318
CSRN (theorem), 319
CSROI (example), 320
CSRST (theorem), 321
CSS (theorem), 322
CSSE (subsection, section CRS), 323
CSSOC (subsection, section CRS), 324
CSTW (example), 325
CTD (subsection, section TD), 326
CTLT (example), 327
CUMOS (theorem), 328
curve fitting
polynomial through 5 points
example PTFP, 329
CV (definition), 330
CV (notation), 331
CV (technique, section PT), 332
CVA (definition), 333
CVA (notation), 334
CVC (notation), 335
CVE (definition), 336
CVE (notation), 337
CVS (example), 338
CVS (subsection, section VR), 339
CVSM (definition), 340
CVSM (example), 341
CVSM (notation), 342
CVSR (example), 343
D (archetype), 344
D (chapter), 345
D (definition), 346
D (notation), 347
D (section), 348
D (subsection, section D), 349
D (subsection, section SD), 350
D (technique, section PT), 351
D33M (example), 352
DAB (example), 353
DC (example), 354
DC (technique, section PT), 355
DC (theorem), 356
DCM (theorem), 357
DCN (Property), 358
DCP (theorem), 359
DD (subsection, section DM), 360
DEC (theorem), 361
decomposition
technique DC, 362
DED (theorem), 363
definition
A, 364
AM, 365
AME, 366
B, 367
C, 368
CBM, 369
CCCV, 370
CCM, 371
CCN, 372
CM, 373
CNA, 374
CNE, 375
CNM, 376
CP, 377
CS, 378
CSM, 379
CV, 380
CVA, 381
CVE, 382
CVSM, 383
D, 384
DIM, 385
DM, 386
DS, 387
DZM, 388
EEF, 389
EELT, 390
EEM, 391
ELEM, 392
EM, 393
EO, 394
ES, 395
ESYS, 396
F, 397
GES, 398
GEV, 399
GME, 400
HI, 401
HID, 402
HM, 403
HP, 404
HS, 405
IDLT, 406
IDV, 407
IE, 408
ILT, 409
IM, 410
IMP, 411
IP, 412
IS, 413
IVLT, 414
IVS, 415
JB, 416
JCF, 417
KLT, 418
LC, 419
LCCV, 420
LI, 421
LICV, 422
LNS, 423
LSMR, 424
LSS, 425
LT, 426
LTA, 427
LTC, 428
LTM, 429
LTR, 430
LTSM, 431
M, 432
MA, 433
MCN, 434
ME, 435
MI, 436
MM, 437
MR, 438
MSM, 439
MVP, 440
NLT, 441
NM, 442
NOLT, 443
NOM, 444
NRML, 445
NSM, 446
NV, 447
ONS, 448
OSV, 449
OV, 450
PI, 451
PSM, 452
REM, 453
RLD, 454
RLDCV, 455
RLT, 456
RO, 457
ROLT, 458
ROM, 459
RR, 460
RREF, 461
RSM, 462
S, 463
SC, 464
SE, 465
SET, 466
SI, 467
SIM, 468
SLE, 469
SLT, 470
SM, 471
SQM, 472
SRM, 473
SS, 474
SSCV, 475
SSET, 476
SU, 477
SUV, 478
SV, 479, 480
SYM, 481
T, 482
technique D, 483
TM, 484
TS, 485
TSHSE, 486
TSVS, 487
UM, 488
UTM, 489
VM, 490
VOC, 491
VR, 492
VS, 493
VSCV, 494
VSM, 495
ZCV, 496
ZM, 497
DEHD (example), 498
DEM (theorem), 499
DEMMM (theorem), 500
DEMS5 (example), 501
DER (theorem), 502
DERC (theorem), 503
determinant
computed two ways
example TCSD, 504
definition DM, 505
equal rows or columns
theorem DERC, 506
expansion, columns
theorem DEC, 507
expansion, rows
theorem DER, 508
identity matrix
theorem DIM, 509
matrix multiplication
theorem DRMM, 510
nonsingular matrix, 511
notation, 512
row or column multiple
theorem DRCM, 513
row or column swap
theorem DRCS, 514
size 2 matrix
theorem DMST, 515
size 3 matrix
example D33M, 516
transpose
theorem DT, 517
via row operations
example DRO, 518
zero
theorem SMZD, 519
zero row or column
theorem DZRC, 520
zero versus nonzero
example ZNDAB, 521
determinant, upper triangular matrix
example DUTM, 522
determinants
elementary matrices
theorem DEMMM, 523
DF (Property), 524
DF (subsection, section CF), 525
DFS (subsection, section PD), 526
DFS (theorem), 527
DGES (theorem), 528
diagonal matrix
definition DIM, 529
diagonalizable
definition DZM, 530
distinct eigenvalues
example DEHD, 531
theorem DED, 532
full eigenspaces
theorem DMFE, 533
not
example NDMS4, 534
diagonalizable matrix
high power
example HPDM, 535
diagonalization
Archetype B
example DAB, 536
criteria
theorem DC, 537
example DMS3, 538
DIM (definition), 539
DIM (theorem), 540
dimension
crazy vector space
example DC, 541
definition D, 542
notation, 543
polynomial subspace
example DSP4, 544
proper subspaces
theorem PSSD, 545
subspace
example DSM22, 546
direct sum
decomposing zero vector
theorem DSZV, 547
definition DS, 548
dimension
theorem DSD, 549
example SDS, 550
from a basis
theorem DSFB, 551
from one subspace
theorem DSFOS, 552
notation, 553
zero intersection
theorem DSZI, 554
direct sums
linear independence
theorem DSLI, 555
repeated
theorem RDS, 556
distributivity
complex numbers
Property DCN, 557
field
Property DF, 558
distributivity, matrix addition
matrices
Property DMAM, 559
distributivity, scalar addition
column vectors
Property DSAC, 560
matrices
Property DSAM, 561
vectors
Property DSA, 562
distributivity, vector addition
column vectors
Property DVAC, 563
vectors
Property DVA, 564
DLDS (theorem), 565
DM (definition), 566
DM (notation), 567
DM (section), 568
DM (theorem), 569
DMAM (Property), 570
DMFE (theorem), 571
DMHP (subsection, section HP), 572
DMHP (theorem), 573
DMMP (theorem), 574
DMS3 (example), 575
DMST (theorem), 576
DNLT (theorem), 577
DNMMM (subsection, section PDM), 578
DP (theorem), 579
DRCM (theorem), 580
DRCMA (theorem), 581
DRCS (theorem), 582
DRMM (theorem), 583
DRO (example), 584
DRO (subsection, section PDM), 585
DROEM (subsection, section PDM), 586
DS (definition), 587
DS (notation), 588
DS (subsection, section PD), 589
DSA (Property), 590
DSAC (Property), 591
DSAM (Property), 592
DSD (theorem), 593
DSFB (theorem), 594
DSFOS (theorem), 595
DSLI (theorem), 596
DSM22 (example), 597
DSP4 (example), 598
DSZI (theorem), 599
DSZV (theorem), 600
DT (theorem), 601
DUTM (example), 602
DVA (Property), 603
DVAC (Property), 604
DVM (theorem), 605
DVS (subsection, section D), 606
DZM (definition), 607
DZRC (theorem), 608
E (archetype), 609
E (chapter), 610
E (technique, section PT), 611
ECEE (subsection, section EE), 612
EDELI (theorem), 613
EDYES (theorem), 614
EE (section), 615
EEE (subsection, section EE), 616
EEF (definition), 617
EEF (subsection, section FS), 618
EELT (definition), 619
EELT (subsection, section CB), 620
EEM (definition), 621
EEM (subsection, section EE), 622
EEMAP (theorem), 623
EENS (example), 624
EER (theorem), 625
EESR (theorem), 626
EHM (subsection, section PEE), 627
eigenspace
as null space
theorem EMNS, 628
definition EM, 629
invariant subspace
theorem EIS, 630
subspace
theorem EMS, 631
eigenvalue
algebraic multiplicity
definition AME, 632
complex
example CEMS6, 633
definition EEM, 634
existence
example CAEHW, 635
theorem EMHE, 636
geometric multiplicity
definition GME, 637
index, 638
linear transformation
definition EELT, 639
multiplicities
example EMMS4, 640
power
theorem EOMP, 641
root of characteristic polynomial
theorem EMRCP, 642
scalar multiple
theorem ESMM, 643
symmetric matrix
example ESMS4, 644
zero
theorem SMZE, 645
eigenvalues
building desired
example BDE, 646
complex, of a linear transformation
example CELT, 647
conjugate pairs
theorem ERMCP, 648
distinct
example DEMS5, 649
example SEE, 650
Hermitian matrices
theorem HMRE, 651
inverse
theorem EIM, 652
maximum number
theorem MNEM, 653
multiplicities
example HMEM5, 654
theorem ME, 655
number
theorem NEM, 656
of a polynomial
theorem EPM, 657
size 3 matrix
example EMS3, 658
example ESMS3, 659
transpose
theorem ETM, 660
eigenvalues, eigenvectors
vector, matrix representations
theorem EER, 661
eigenvector, 662
linear transformation, 663
eigenvectors, 664
conjugate pairs, 665
Hermitian matrices
theorem HMOE, 666
linear transformation
example ELTBM, 667
example ELTBP, 668
linearly independent
theorem EDELI, 669
of a linear transformation
example ELTT, 670
EILT (subsection, section ILT), 671
EIM (theorem), 672
EIS (example), 673
EIS (theorem), 674
ELEM (definition), 675
ELEM (notation), 676
elementary matrices
definition ELEM, 677
determinants
theorem DEM, 678
nonsingular
theorem EMN, 679
notation, 680
row operations
example EMRO, 681
theorem EMDRO, 682
ELIS (theorem), 683
ELTBM (example), 684
ELTBP (example), 685
ELTT (example), 686
EM (definition), 687
EM (subsection, section DM), 688
EMDRO (theorem), 689
EMHE (theorem), 690
EMMS4 (example), 691
EMMVP (theorem), 692
EMN (theorem), 693
EMNS (theorem), 694
EMP (theorem), 695
empty set, 696
notation, 697
EMRCP (theorem), 698
EMRO (example), 699
EMS (theorem), 700
EMS3 (example), 701
ENLT (theorem), 702
EO (definition), 703
EOMP (theorem), 704
EOPSS (theorem), 705
EPM (theorem), 706
EPSM (theorem), 707
equal matrices
via equal matrix-vector products
theorem EMMVP, 708
equation operations
definition EO, 709
theorem EOPSS, 710
equivalence statements
technique E, 711
equivalences
technique ME, 712
equivalent systems
definition ESYS, 713
ERMCP (theorem), 714
ES (definition), 715
ES (notation), 716
ESEO (subsection, section SSLE), 717
ESLT (subsection, section SLT), 718
ESMM (theorem), 719
ESMS3 (example), 720
ESMS4 (example), 721
ESYS (definition), 722
ETM (theorem), 723
EVS (subsection, section VS), 724
example
AALC, 725
ABLC, 726
ABS, 727
ACN, 728
AHSAC, 729
AIVLT, 730
ALT, 731
ALTMM, 732
AM, 733
AMAA, 734
ANILT, 735
ANM, 736
AOS, 737
ASC, 738
AVR, 739
BC, 740
BDE, 741
BDM22, 742
BM, 743
BP, 744
BPR, 745
BRLT, 746
BSM22, 747
BSP4, 748
CABAK, 749
CAEHW, 750
CBCV, 751
CBP, 752
CCM, 753
CELT, 754
CEMS6, 755
CFNLT, 756
CFV, 757
CM32, 758
CMI, 759
CMIAB, 760
CNS1, 761
CNS2, 762
CNSV, 763
COV, 764
CP2, 765
CPMS3, 766
CROB3, 767
CROB4, 768
CS, 769
CSAA, 770
CSAB, 771
CSANS, 772
CSCN, 773
CSIP, 774
CSMCS, 775
CSOCD, 776
CSROI, 777
CSTW, 778
CTLT, 779
CVS, 780
CVSM, 781
CVSR, 782
D33M, 783
DAB, 784
DC, 785
DEHD, 786
DEMS5, 787
DMS3, 788
DRO, 789
DSM22, 790
DSP4, 791
DUTM, 792
EENS, 793
EIS, 794
ELTBM, 795
ELTBP, 796
ELTT, 797
EMMS4, 798
EMRO, 799
EMS3, 800
ESMS3, 801
ESMS4, 802
FDV, 803
FF8, 804
FRAN, 805
FS1, 806
FS2, 807
FSAG, 808
GE4, 809
GE6, 810
GENR6, 811
GSTV, 812
HISAA, 813
HISAD, 814
HMEM5, 815
HP, 816
HPDM, 817
HUSAB, 818
IAP, 819
IAR, 820
IAS, 821
IAV, 822
ILTVR, 823
IM, 824
IM11, 825
IS, 826
ISJB, 827
ISMR4, 828
ISMR6, 829
ISSI, 830
IVSAV, 831
JB4, 832
JCF10, 833
KPNLT, 834
KVMR, 835
LCM, 836
LDCAA, 837
LDHS, 838
LDP4, 839
LDRN, 840
LDS, 841
LIC, 842
LICAB, 843
LIHS, 844
LIM32, 845
LINSB, 846
LIP4, 847
LIS, 848
LLDS, 849
LNS, 850
LTDB1, 851
LTDB2, 852
LTDB3, 853
LTM, 854
LTPM, 855
LTPP, 856
LTRGE, 857
MA, 858
MBC, 859
MCSM, 860
MFLT, 861
MI, 862
MIVS, 863
MMNC, 864
MNSLE, 865
MOLT, 866
MPMR, 867
MRBE, 868
MRCM, 869
MSCN, 870
MSM, 871
MTV, 872
MWIAA, 873
NDMS4, 874
NIAO, 875
NIAQ, 876
NIAQR, 877
NIDAU, 878
NJB5, 879
NKAO, 880
NLT, 881
NM, 882
NM62, 883
NM64, 884
NM83, 885
NRREF, 886
NSAO, 887
NSAQ, 888
NSAQR, 889
NSC2A, 890
NSC2S, 891
NSC2Z, 892
NSDAT, 893
NSDS, 894
NSE, 895
NSEAI, 896
NSLE, 897
NSLIL, 898
NSNM, 899
NSR, 900
NSS, 901
OLTTR, 902
ONFV, 903
ONTV, 904
OSGMD, 905
OSMC, 906
PCVS, 907
PM, 908
PSHS, 909
PTFP, 910
PTM, 911
PTMEE, 912
RAO, 913
RES, 914
RNM, 915
RNSM, 916
ROD2, 917
ROD4, 918
RREF, 919
RREFN, 920
RRTI, 921
RS, 922
RSAI, 923
RSB, 924
RSC5, 925
RSNS, 926
RSREM, 927
RSSC4, 928
RVMR, 929
S, 930
SAA, 931
SAB, 932
SABMI, 933
SAE, 934
SAN, 935
SAR, 936
SAV, 937
SC, 938
SC3, 939
SCAA, 940
SCAB, 941
SCAD, 942
SDS, 943
SEE, 944
SEEF, 945
SETM, 946
SI, 947
SM2Z7, 948
SM32, 949
SMLT, 950
SMS3, 951
SMS5, 952
SP4, 953
SPIAS, 954
SRR, 955
SS, 956
SS6W, 957
SSC, 958
SSET, 959
SSM22, 960
SSNS, 961
SSP, 962
SSP4, 963
STLT, 964
STNE, 965
SU, 966
SUVOS, 967
SVP4, 968
SYM, 969
TCSD, 970
TD4, 971
TDEE6, 972
TDSSE, 973
TIS, 974
TIVS, 975
TKAP, 976
TLC, 977
TM, 978
TMP, 979
TOV, 980
TREM, 981
TTS, 982
UM3, 983
UPM, 984
US, 985
USR, 986
VA, 987
VESE, 988
VFS, 989
VFSAD, 990
VFSAI, 991
VFSAL, 992
VM4, 993
VRC4, 994
VRP2, 995
VSCV, 996
VSF, 997
VSIM5, 998
VSIS, 999
VSM, 1000
VSP, 1001
VSPUD, 1002
VSS, 1003
ZNDAB, 1004
EXC (subsection, section B), 1005
EXC (subsection, section CB), 1006
EXC (subsection, section CF), 1007
EXC (subsection, section CRS), 1008
EXC (subsection, section D), 1009
EXC (subsection, section DM), 1010
EXC (subsection, section EE), 1011
EXC (subsection, section F), 1012
EXC (subsection, section FS), 1013
EXC (subsection, section HP), 1014
EXC (subsection, section HSE), 1015
EXC (subsection, section ILT), 1016
EXC (subsection, section IVLT), 1017
EXC (subsection, section LC), 1018
EXC (subsection, section LDS), 1019
EXC (subsection, section LI), 1020
EXC (subsection, section LISS), 1021
EXC (subsection, section LT), 1022
EXC (subsection, section MINM), 1023
EXC (subsection, section MISLE), 1024
EXC (subsection, section MM), 1025
EXC (subsection, section MO), 1026
EXC (subsection, section MR), 1027
EXC (subsection, section NM), 1028
EXC (subsection, section O), 1029
EXC (subsection, section PD), 1030
EXC (subsection, section PDM), 1031
EXC (subsection, section PEE), 1032
EXC (subsection, section PSM), 1033
EXC (subsection, section RREF), 1034
EXC (subsection, section S), 1035
EXC (subsection, section SD), 1036
EXC (subsection, section SLT), 1037
EXC (subsection, section SS), 1038
EXC (subsection, section SSLE), 1039
EXC (subsection, section T), 1040
EXC (subsection, section TSS), 1041
EXC (subsection, section VO), 1042
EXC (subsection, section VR), 1043
EXC (subsection, section VS), 1044
EXC (subsection, section WILA), 1045
extended echelon form
submatrices
example SEEF, 1046
extended reduced row-echelon form
properties
theorem PEEF, 1047
F (archetype), 1048
F (definition), 1049
F (section), 1050
F (subsection, section F), 1051
FDV (example), 1052
FF (subsection, section F), 1053
FF8 (example), 1054
field
definition F, 1055
FIMP (theorem), 1056
finite field
size 8
example FF8, 1057
four subsets
example FS1, 1058
example FS2, 1059
four subspaces
dimension
theorem DFS, 1060
FRAN (example), 1061
free variables
example CFV, 1062
free variables, number
theorem FVCS, 1063
free, independent variables
example FDV, 1064
FS (section), 1065
FS (subsection, section FS), 1066
FS (theorem), 1067
FS1 (example), 1068
FS2 (example), 1069
FSAG (example), 1070
FTMR (theorem), 1071
FV (subsection, section TSS), 1072
FVCS (theorem), 1073
G (archetype), 1074
G (theorem), 1075
GE4 (example), 1076
GE6 (example), 1077
GEE (subsection, section IS), 1078
GEK (theorem), 1079
generalized eigenspace
as kernel
theorem GEK, 1080
definition GES, 1081
dimension
theorem DGES, 1082
dimension 4 domain
example GE4, 1083
dimension 6 domain
example GE6, 1084
invariant subspace
theorem GESIS, 1085
nilpotent restriction
theorem RGEN, 1086
nilpotent restrictions, dimension 6 domain
example GENR6, 1087
notation, 1088
generalized eigenspace decomposition
theorem GESD, 1089
generalized eigenvector
definition GEV, 1090
GENR6 (example), 1091
GES (definition), 1092
GES (notation), 1093
GESD (subsection, section JCF), 1094
GESD (theorem), 1095
GESIS (theorem), 1096
GEV (definition), 1097
GFDL (appendix), 1098
GME (definition), 1099
goldilocks
theorem G, 1100
Gram-Schmidt
column vectors
theorem GSP, 1101
three vectors
example GSTV, 1102
gram-schmidt
mathematica, 1103
GS (technique, section PT), 1104
GSP (subsection, section O), 1105
GSP (theorem), 1106
GSP.MMA (computation, section MMA), 1107
GSTV (example), 1108
GT (subsection, section PD), 1109
H (archetype), 1110
Hadamard Identity
notation, 1111
Hadamard identity
definition HID, 1112
Hadamard Inverse
notation, 1113
Hadamard inverse
definition HI, 1114
Hadamard Product
Diagonalizable Matrices
theorem DMHP, 1115
notation, 1116
Hadamard product
commutativity
theorem HPC, 1117
definition HP, 1118
diagonal matrices
theorem DMMP, 1119
distributivity
theorem HPDAA, 1120
example HP, 1121
identity
theorem HPHID, 1122
inverse
theorem HPHI, 1123
scalar matrix multiplication
theorem HPSMM, 1124
hermitian
definition HM, 1125
Hermitian matrix
inner product
theorem HMIP, 1126
HI (definition), 1127
HI (notation), 1128
HID (definition), 1129
HID (notation), 1130
HISAA (example), 1131
HISAD (example), 1132
HM (definition), 1133
HM (subsection, section MM), 1134
HMEM5 (example), 1135
HMIP (theorem), 1136
HMOE (theorem), 1137
HMRE (theorem), 1138
HMVEI (theorem), 1139
homogeneous system
consistent
theorem HSC, 1140
definition HS, 1141
infinitely many solutions
theorem HMVEI, 1142
homogeneous systems
linear independence, 1143
homogenous system
Archetype C
example AHSAC, 1144
HP (definition), 1145
HP (example), 1146
HP (notation), 1147
HP (section), 1148
HPC (theorem), 1149
HPDAA (theorem), 1150
HPDM (example), 1151
HPHI (theorem), 1152
HPHID (theorem), 1153
HPSMM (theorem), 1154
HS (definition), 1155
HSC (theorem), 1156
HSE (section), 1157
HUSAB (example), 1158
I (archetype), 1159
I (technique, section PT), 1160
IAP (example), 1161
IAR (example), 1162
IAS (example), 1163
IAV (example), 1164
ICBM (theorem), 1165
ICLT (theorem), 1166
identities
technique PI, 1167
identity matrix
determinant, 1168
example IM, 1169
notation, 1170
IDLT (definition), 1171
IDV (definition), 1172
IE (definition), 1173
IE (notation), 1174
IFDVS (theorem), 1175
IILT (theorem), 1176
ILT (definition), 1177
ILT (section), 1178
ILTB (theorem), 1179
ILTD (subsection, section ILT), 1180
ILTD (theorem), 1181
ILTIS (theorem), 1182
ILTLI (subsection, section ILT), 1183
ILTLI (theorem), 1184
ILTLT (theorem), 1185
ILTVR (example), 1186
IM (definition), 1187
IM (example), 1188
IM (notation), 1189
IM (subsection, section MISLE), 1190
IM11 (example), 1191
IMILT (theorem), 1192
IMP (definition), 1193
IMR (theorem), 1194
inconsistent linear systems
theorem ISRN, 1195
independent, dependent variables
definition IDV, 1196
indesxstring
example SM2Z7, 1197
example SSET, 1198
index
eigenvalue
definition IE, 1199
notation, 1200
indexstring
theorem CBOB, 1201
theorem DRCMA, 1202
theorem OBUTR, 1203
theorem UMCOB, 1204
induction
technique I, 1205
infinite solution set
example ISSI, 1206
infinite solutions,
example IS, 1207
injective
example IAP, 1208
example IAR, 1209
not
example NIAO, 1210
example NIAQ, 1211
example NIAQR, 1212
not, by dimension
example NIDAU, 1213
polynomials to matrices
example IAV, 1214
injective linear transformation
bases
theorem ILTB, 1215
injective linear transformations
dimension
theorem ILTD, 1216
inner product
anti-commutative
theorem IPAC, 1217
example CSIP, 1218
norm
theorem IPN, 1219
notation, 1220
positive
theorem PIP, 1221
scalar multiplication
theorem IPSM, 1222
vector addition
theorem IPVA, 1223
integers
mod
definition IMP, 1224
mod , field
theorem FIMP, 1225
mod 11
example IM11, 1226
interpolating polynomial
theorem IP, 1227
invariant subspace
definition IS, 1228
eigenspace, 1229
eigenspaces
example EIS, 1230
example TIS, 1231
Jordan block
example ISJB, 1232
kernels of powers
theorem KPIS, 1233
inverse
composition of linear transformations
theorem ICLT, 1234
example CMI, 1235
example MI, 1236
notation, 1237
of a matrix, 1238
invertible linear transformation
defined by invertible matrix
theorem IMILT, 1239
invertible linear transformations
composition
theorem CIVLT, 1240
IP (definition), 1241
IP (notation), 1242
IP (subsection, section O), 1243
IP (theorem), 1244
IPAC (theorem), 1245
IPN (theorem), 1246
IPSM (theorem), 1247
IPVA (theorem), 1248
IS (definition), 1249
IS (example), 1250
IS (section), 1251
IS (subsection, section IS), 1252
ISJB (example), 1253
ISMR4 (example), 1254
ISMR6 (example), 1255
isomorphic
multiple vector spaces
example MIVS, 1256
vector spaces
example IVSAV, 1257
isomorphic vector spaces
dimension
theorem IVSED, 1258
example TIVS, 1259
ISRN (theorem), 1260
ISSI (example), 1261
ITMT (theorem), 1262
IV (subsection, section IVLT), 1263
IVLT (definition), 1264
IVLT (section), 1265
IVLT (subsection, section IVLT), 1266
IVLT (subsection, section MR), 1267
IVS (definition), 1268
IVSAV (example), 1269
IVSED (theorem), 1270
J (archetype), 1271
JB (definition), 1272
JB (notation), 1273
JB4 (example), 1274
JCF (definition), 1275
JCF (section), 1276
JCF (subsection, section JCF), 1277
JCF10 (example), 1278
JCFLT (theorem), 1279
Jordan block
definition JB, 1280
nilpotent
theorem NJB, 1281
notation, 1282
size 4
example JB4, 1283
Jordan canonical form
definition JCF, 1284
size 10
example JCF10, 1285
K (archetype), 1286
kernel
injective linear transformation
theorem KILT, 1287
isomorphic to null space
theorem KNSI, 1288
linear transformation
example NKAO, 1289
notation, 1290
of a linear transformation
definition KLT, 1291
pre-image, 1292
subspace
theorem KLTS, 1293
trivial
example TKAP, 1294
via matrix representation
example KVMR, 1295
KILT (theorem), 1296
KLT (definition), 1297
KLT (notation), 1298
KLT (subsection, section ILT), 1299
KLTS (theorem), 1300
KNSI (theorem), 1301
KPI (theorem), 1302
KPIS (theorem), 1303
KPLT (theorem), 1304
KPNLT (example), 1305
KPNLT (theorem), 1306
KVMR (example), 1307
L (archetype), 1308
L (technique, section PT), 1309
LA (subsection, section WILA), 1310
LC (definition), 1311
LC (section), 1312
LC (subsection, section LC), 1313
LC (technique, section PT), 1314
LCCV (definition), 1315
LCM (example), 1316
LDCAA (example), 1317
LDHS (example), 1318
LDP4 (example), 1319
LDRN (example), 1320
LDS (example), 1321
LDS (section), 1322
LDSS (subsection, section LDS), 1323
least squares
minimizes residuals
theorem LSMR, 1324
least squares solution
definition LSS, 1325
left null space
as row space, 1326
definition LNS, 1327
example LNS, 1328
notation, 1329
subspace
theorem LNSMS, 1330
lemma
technique LC, 1331
LI (definition), 1332
LI (section), 1333
LI (subsection, section LISS), 1334
LIC (example), 1335
LICAB (example), 1336
LICV (definition), 1337
LIHS (example), 1338
LIM32 (example), 1339
linear combination
system of equations
example ABLC, 1340
definition LC, 1341
definition LCCV, 1342
example TLC, 1343
linear transformation, 1344
matrices
example LCM, 1345
system of equations
example AALC, 1346
linear combinations
solutions to linear systems
theorem SLSLC, 1347
linear dependence
more vectors than size
theorem MVSLD, 1348
linear independence
definition LI, 1349
definition LICV, 1350
homogeneous systems
theorem LIVHS, 1351
injective linear transformation
theorem ILTLI, 1352
matrices
example LIM32, 1353
orthogonal, 1354
r and n
theorem LIVRN, 1355
linear solve
mathematica, 1356
linear system
consistent
theorem RCLS, 1357
matrix representation
definition LSMR, 1358
notation, 1359
linear systems
notation
example MNSLE, 1360
example NSLE, 1361
linear transformation
polynomials to polynomials
example LTPP, 1362
addition
definition LTA, 1363
theorem MLTLT, 1364
theorem SLTLT, 1365
as matrix multiplication
example ALTMM, 1366
basis of range
example BRLT, 1367
checking
example ALT, 1368
composition
definition LTC, 1369
theorem CLTLT, 1370
defined by a matrix
example LTM, 1371
defined on a basis
example LTDB1, 1372
example LTDB2, 1373
example LTDB3, 1374
theorem LTDB, 1375
definition LT, 1376
identity
definition IDLT, 1377
injection
definition ILT, 1378
inverse
theorem ILTLT, 1379
inverse of inverse
theorem IILT, 1380
invertible
definition IVLT, 1381
example AIVLT, 1382
invertible, injective and surjective
theorem ILTIS, 1383
Jordan canonical form
theorem JCFLT, 1384
kernels of powers
theorem KPLT, 1385
linear combination
theorem LTLC, 1386
matrix of, 1387
example MFLT, 1388
example MOLT, 1389
not
example NLT, 1390
not invertible
example ANILT, 1391
notation, 1392
polynomials to matrices
example LTPM, 1393
rank plus nullity
theorem RPNDD, 1394
restriction
definition LTR, 1395
notation, 1396
scalar multiple
example SMLT, 1397
scalar multiplication
definition LTSM, 1398
spanning range
theorem SSRLT, 1399
sum
example STLT, 1400
surjection
definition SLT, 1401
vector space of, 1402
zero vector
theorem LTTZZ, 1403
linear transformation inverse
via matrix representation
example ILTVR, 1404
linear transformation restriction
on generalized eigenspace
example LTRGE, 1405
linear transformations
compositions
example CTLT, 1406
from matrices
theorem MBLT, 1407
linearly dependent
example LDRN, 1408
via homogeneous system
example LDHS, 1409
linearly dependent columns
Archetype A
example LDCAA, 1410
linearly dependent set
example LDS, 1411
linear combinations within
theorem DLDS, 1412
polynomials
example LDP4, 1413
linearly independent
crazy vector space
example LIC, 1414
extending sets
theorem ELIS, 1415
polynomials
example LIP4, 1416
via homogeneous system
example LIHS, 1417
linearly independent columns
Archetype B
example LICAB, 1418
linearly independent set
example LIS, 1419
example LLDS, 1420
LINM (subsection, section LI), 1421
LINSB (example), 1422
LIP4 (example), 1423
LIS (example), 1424
LISS (section), 1425
LISV (subsection, section LI), 1426
LIVHS (theorem), 1427
LIVRN (theorem), 1428
LLDS (example), 1429
LNS (definition), 1430
LNS (example), 1431
LNS (notation), 1432
LNS (subsection, section FS), 1433
LNSMS (theorem), 1434
lower triangular matrix
definition LTM, 1435
LS.MMA (computation, section MMA), 1436
LSMR (definition), 1437
LSMR (notation), 1438
LSMR (theorem), 1439
LSS (definition), 1440
LT (chapter), 1441
LT (definition), 1442
LT (notation), 1443
LT (section), 1444
LT (subsection, section LT), 1445
LTA (definition), 1446
LTC (definition), 1447
LTDB (theorem), 1448
LTDB1 (example), 1449
LTDB2 (example), 1450
LTDB3 (example), 1451
LTLC (subsection, section LT), 1452
LTLC (theorem), 1453
LTM (definition), 1454
LTM (example), 1455
LTPM (example), 1456
LTPP (example), 1457
LTR (definition), 1458
LTR (notation), 1459
LTRGE (example), 1460
LTSM (definition), 1461
LTTZZ (theorem), 1462
M (archetype), 1463
M (chapter), 1464
M (definition), 1465
M (notation), 1466
MA (definition), 1467
MA (example), 1468
MA (notation), 1469
MACN (Property), 1470
MAF (Property), 1471
MAP (subsection, section SVD), 1472
mathematica
gram-schmidt (computation), 1473
linear solve (computation), 1474
matrix entry (computation), 1475
matrix inverse (computation), 1476
matrix multiplication (computation), 1477
null space (computation), 1478
row reduce (computation), 1479
transpose of a matrix (computation), 1480
vector form of solutions (computation), 1481
vector linear combinations (computation), 1482
mathematical language
technique L, 1483
matrix
addition
definition MA, 1484
notation, 1485
augmented
definition AM, 1486
column space
definition CSM, 1487
complex conjugate
example CCM, 1488
definition M, 1489
equality
definition ME, 1490
notation, 1491
example AM, 1492
identity
definition IM, 1493
inverse
definition MI, 1494
nonsingular
definition NM, 1495
notation, 1496
of a linear transformation
theorem MLTCV, 1497
product
example PTM, 1498
example PTMEE, 1499
product with vector
definition MVP, 1500
rectangular, 1501
row space
definition RSM, 1502
scalar multiplication
definition MSM, 1503
notation, 1504
singular, 1505
square
definition SQM, 1506
submatrices
example SS, 1507
submatrix
definition SM, 1508
symmetric
definition SYM, 1509
transpose
definition TM, 1510
unitary
definition UM, 1511
unitary is invertible
theorem UMI, 1512
zero
definition ZM, 1513
matrix addition
example MA, 1514
matrix components
notation, 1515
matrix entry
mathematica, 1516
ti83, 1517
ti86, 1518
matrix inverse
Archetype B, 1519
computation
theorem CINM, 1520
mathematica, 1521
nonsingular matrix
theorem NI, 1522
of a matrix inverse
theorem MIMI, 1523
one-sided
theorem OSIS, 1524
product
theorem SS, 1525
scalar multiple
theorem MISM, 1526
size 2 matrices
theorem TTMI, 1527
transpose
theorem MIT, 1528
uniqueness
theorem MIU, 1529
matrix multiplication
adjoints
theorem MMAD, 1530
associativity
theorem MMA, 1531
complex conjugation
theorem MMCC, 1532
definition MM, 1533
distributivity
theorem MMDAA, 1534
entry-by-entry
theorem EMP, 1535
identity matrix
theorem MMIM, 1536
inner product
theorem MMIP, 1537
mathematica, 1538
noncommutative
example MMNC, 1539
scalar matrix multiplication
theorem MMSMM, 1540
systems of linear equations
theorem SLEMM, 1541
transposes
theorem MMT, 1542
zero matrix
theorem MMZM, 1543
matrix product
as composition of linear transformations
example MPMR, 1544
matrix representation
basis of eigenvectors
example MRBE, 1545
composition of linear transformations
theorem MRCLT, 1546
definition MR, 1547
invertible
theorem IMR, 1548
multiple of a linear transformation
theorem MRMLT, 1549
restriction to generalized eigenspace
theorem MRRGE, 1550
sum of linear transformations
theorem MRSLT, 1551
theorem FTMR, 1552
upper triangular
theorem UTMR, 1553
matrix representations
converting with change-of-basis
example MRCM, 1554
example OLTTR, 1555
matrix scalar multiplication
example MSM, 1556
matrix vector space
dimension
theorem DM, 1557
matrix-adjoint product
eigenvalues, eigenvectors
theorem EEMAP, 1558
matrix-vector product
example MTV, 1559
notation, 1560
MBC (example), 1561
MBLT (theorem), 1562
MC (notation), 1563
MCC (subsection, section MO), 1564
MCCN (Property), 1565
MCF (Property), 1566, 1567
MCN (definition), 1568
MCN (subsection, section CNO), 1569
MCSM (example), 1570
MCT (theorem), 1571
MD (chapter), 1572
ME (definition), 1573
ME (notation), 1574
ME (subsection, section PEE), 1575
ME (technique, section PT), 1576
ME (theorem), 1577
ME.MMA (computation, section MMA), 1578
ME.TI83 (computation, section TI83), 1579
ME.TI86 (computation, section TI86), 1580
MEASM (subsection, section MO), 1581
MFLT (example), 1582
MI (definition), 1583
MI (example), 1584
MI (notation), 1585
MI.MMA (computation, section MMA), 1586
MICN (Property), 1587
MIF (Property), 1588
MIMI (theorem), 1589
MINM (section), 1590
MISLE (section), 1591
MISM (theorem), 1592
MIT (theorem), 1593
MIU (theorem), 1594
MIVS (example), 1595
MLT (subsection, section LT), 1596
MLTCV (theorem), 1597
MLTLT (theorem), 1598
MM (definition), 1599
MM (section), 1600
MM (subsection, section MM), 1601
MM.MMA (computation, section MMA), 1602
MMA (section), 1603
MMA (theorem), 1604
MMAD (theorem), 1605
MMCC (theorem), 1606
MMDAA (theorem), 1607
MMEE (subsection, section MM), 1608
MMIM (theorem), 1609
MMIP (theorem), 1610
MMNC (example), 1611
MMSMM (theorem), 1612
MMT (theorem), 1613
MMZM (theorem), 1614
MNEM (theorem), 1615
MNSLE (example), 1616
MO (section), 1617
MOLT (example), 1618
more variables than equations
example OSGMD, 1619
theorem CMVEI, 1620
MPMR (example), 1621
MR (definition), 1622
MR (section), 1623
MRBE (example), 1624
MRCB (theorem), 1625
MRCLT (theorem), 1626
MRCM (example), 1627
MRMLT (theorem), 1628
MRRGE (theorem), 1629
MRS (subsection, section CB), 1630
MRSLT (theorem), 1631
MSCN (example), 1632
MSM (definition), 1633
MSM (example), 1634
MSM (notation), 1635
MTV (example), 1636
multiplicative associativity
complex numbers
Property MACN, 1637
multiplicative closure
field
Property MCF, 1638
multiplicative commuativity
complex numbers
Property MCCN, 1639
multiplicative inverse
complex numbers
Property MICN, 1640
MVNSE (subsection, section RREF), 1641
MVP (definition), 1642
MVP (notation), 1643
MVP (subsection, section MM), 1644
MVSLD (theorem), 1645
MWIAA (example), 1646
N (archetype), 1647
N (subsection, section O), 1648
N (technique, section PT), 1649
NDMS4 (example), 1650
negation of statements
technique N, 1651
NEM (theorem), 1652
NI (theorem), 1653
NIAO (example), 1654
NIAQ (example), 1655
NIAQR (example), 1656
NIDAU (example), 1657
nilpotent
linear transformation
definition NLT, 1658
NJB (theorem), 1659
NJB5 (example), 1660
NKAO (example), 1661
NLT (definition), 1662
NLT (example), 1663
NLT (section), 1664
NLT (subsection, section NLT), 1665
NLTFO (subsection, section LT), 1666
NM (definition), 1667
NM (example), 1668
NM (section), 1669
NM (subsection, section NM), 1670
NM (subsection, section OD), 1671
NM62 (example), 1672
NM64 (example), 1673
NM83 (example), 1674
NME1 (theorem), 1675
NME2 (theorem), 1676
NME3 (theorem), 1677
NME4 (theorem), 1678
NME5 (theorem), 1679
NME6 (theorem), 1680
NME7 (theorem), 1681
NME8 (theorem), 1682
NME9 (theorem), 1683
NMI (subsection, section MINM), 1684
NMLIC (theorem), 1685
NMPEM (theorem), 1686
NMRRI (theorem), 1687
NMTNS (theorem), 1688
NMUS (theorem), 1689
NOILT (theorem), 1690
NOLT (definition), 1691
NOLT (notation), 1692
NOM (definition), 1693
NOM (notation), 1694
nonsingular
columns as basis
theorem CNMB, 1695
nonsingular matrices
linearly independent columns
theorem NMLIC, 1696
nonsingular matrix
Archetype B
example NM, 1697
column space, 1698
elemntary matrices
theorem NMPEM, 1699
equivalences
theorem NME1, 1700
theorem NME2, 1701
theorem NME3, 1702
theorem NME4, 1703
theorem NME5, 1704
theorem NME6, 1705
theorem NME7, 1706
theorem NME8, 1707
theorem NME9, 1708
matrix inverse, 1709
null space
example NSNM, 1710
nullity, 1711
product of nonsingular matrices
theorem NPNT, 1712
rank
theorem RNNM, 1713
row-reduced
theorem NMRRI, 1714
trivial null space
theorem NMTNS, 1715
unique solutions
theorem NMUS, 1716
nonsingular matrix, row-reduced
example NSR, 1717
norm
example CNSV, 1718
inner product, 1719
notation, 1720
normal matrix
definition NRML, 1721
example ANM, 1722
orthonormal basis, 1723
notation
A, 1724
AM, 1725
C, 1726
CCCV, 1727
CCM, 1728
CCN, 1729
CNA, 1730
CNE, 1731
CNM, 1732
CSM, 1733
CV, 1734
CVA, 1735
CVC, 1736
CVE, 1737
CVSM, 1738
D, 1739
DM, 1740
DS, 1741
ELEM, 1742
ES, 1743
GES, 1744
HI, 1745
HID, 1746
HP, 1747
IE, 1748
IM, 1749
IP, 1750
JB, 1751
KLT, 1752
LNS, 1753
LSMR, 1754
LT, 1755
LTR, 1756
M, 1757
MA, 1758
MC, 1759
ME, 1760
MI, 1761
MSM, 1762
MVP, 1763
NOLT, 1764
NOM, 1765
NSM, 1766
NV, 1767
RLT, 1768
RO, 1769
ROLT, 1770
ROM, 1771
RREFA, 1772
RSM, 1773
SC, 1774
SE, 1775
SETM, 1776
SI, 1777
SM, 1778
SRM, 1779
SSET, 1780
SSV, 1781
SU, 1782
T, 1783
TM, 1784
VSCV, 1785
VSM, 1786
ZCV, 1787
ZM, 1788
notation for a linear system
example NSE, 1789
NPNT (theorem), 1790
NRFO (subsection, section MR), 1791
NRML (definition), 1792
NRREF (example), 1793
NS.MMA (computation, section MMA), 1794
NSAO (example), 1795
NSAQ (example), 1796
NSAQR (example), 1797
NSC2A (example), 1798
NSC2S (example), 1799
NSC2Z (example), 1800
NSDAT (example), 1801
NSDS (example), 1802
NSE (example), 1803
NSEAI (example), 1804
NSLE (example), 1805
NSLIL (example), 1806
NSM (definition), 1807
NSM (notation), 1808
NSM (subsection, section HSE), 1809
NSMS (theorem), 1810
NSNM (example), 1811
NSNM (subsection, section NM), 1812
NSR (example), 1813
NSS (example), 1814
NSSLI (subsection, section LI), 1815
Null space
as a span
example NSDS, 1816
null space
Archetype I
example NSEAI, 1817
basis
theorem BNS, 1818
computation
example CNS1, 1819
example CNS2, 1820
isomorphic to kernel, 1821
linearly independent basis
example LINSB, 1822
mathematica, 1823
matrix
definition NSM, 1824
nonsingular matrix, 1825
notation, 1826
singular matrix, 1827
spanning set
example SSNS, 1828
theorem SSNS, 1829
subspace
theorem NSMS, 1830
null space span, linearly independent
Archetype L
example NSLIL, 1831
nullity
computing, 1832
injective linear transformation
theorem NOILT, 1833
linear transformation
definition NOLT, 1834
matrix, 1835
definition NOM, 1836
notation, 1837, 1838
square matrix, 1839
NV (definition), 1840
NV (notation), 1841
NVM (theorem), 1842
O (archetype), 1843
O (Property), 1844
O (section), 1845
OBC (subsection, section B), 1846
OBNM (theorem), 1847
OBUTR (theorem), 1848
OC (Property), 1849
OCN (Property), 1850
OD (section), 1851
OD (subsection, section OD), 1852
OD (theorem), 1853
OF (Property), 1854
OLTTR (example), 1855
OM (Property), 1856
one
column vectors
Property OC, 1857
complex numbers
Property OCN, 1858
field
Property OF, 1859
matrices
Property OM, 1860
vectors
Property O, 1861
ONFV (example), 1862
ONS (definition), 1863
ONTV (example), 1864
orthogonal
linear independence
theorem OSLI, 1865
set
example AOS, 1866
set of vectors
definition OSV, 1867
vector pairs
definition OV, 1868
orthogonal vectors
example TOV, 1869
orthonormal
definition ONS, 1870
matrix columns
example OSMC, 1871
orthonormal basis
normal matrix
theorem OBNM, 1872
orthonormal diagonalization
theorem OD, 1873
orthonormal set
four vectors
example ONFV, 1874
three vectors
example ONTV, 1875
OSGMD (example), 1876
OSIS (theorem), 1877
OSLI (theorem), 1878
OSMC (example), 1879
OSV (definition), 1880
OV (definition), 1881
OV (subsection, section O), 1882
P (appendix), 1883
P (archetype), 1884
P (technique, section PT), 1885
particular solutions
example PSHS, 1886
PCNA (theorem), 1887
PCVS (example), 1888
PD (section), 1889
PDM (section), 1890
PDM (theorem), 1891
PEE (section), 1892
PEEF (theorem), 1893
PI (definition), 1894
PI (subsection, section LT), 1895
PI (technique, section PT), 1896
PIP (theorem), 1897
PM (example), 1898
PM (subsection, section EE), 1899
PMI (subsection, section MISLE), 1900
PMM (subsection, section MM), 1901
PMR (subsection, section MR), 1902
PNLT (subsection, section NLT), 1903
POD (section), 1904
polar decomposition
theorem PDM, 1905
polynomial
of a matrix
example PM, 1906
polynomial vector space
dimension
theorem DP, 1907
positive semi-definite
creating
theorem CPSM, 1908
positive semi-definite matrix
definition PSM, 1909
eigenvalues
theorem EPSM, 1910
practice
technique P, 1911
pre-image
definition PI, 1912
kernel
theorem KPI, 1913
pre-images
example SPIAS, 1914
principal axis theorem, 1915
product of triangular matrices
theorem PTMT, 1916
Property
AA, 1917
AAC, 1918
AACN, 1919
AAF, 1920
AAM, 1921
AC, 1922
ACC, 1923
ACCN, 1924
ACF, 1925, 1926
ACM, 1927
AI, 1928
AIC, 1929
AICN, 1930
AIF, 1931
AIM, 1932
C, 1933
CC, 1934
CM, 1935
DCN, 1936
DF, 1937
DMAM, 1938
DSA, 1939
DSAC, 1940
DSAM, 1941
DVA, 1942
DVAC, 1943
MACN, 1944
MAF, 1945
MCCN, 1946
MCF, 1947, 1948
MICN, 1949
MIF, 1950
O, 1951
OC, 1952
OCN, 1953
OF, 1954
OM, 1955
SC, 1956
SCC, 1957
SCM, 1958
SMA, 1959
SMAC, 1960
SMAM, 1961
Z, 1962
ZC, 1963
ZCN, 1964
ZF, 1965
ZM, 1966
PSHS (example), 1967
PSHS (subsection, section LC), 1968
PSM (definition), 1969
PSM (section), 1970
PSM (subsection, section PSM), 1971
PSM (subsection, section SD), 1972
PSMSR (theorem), 1973
PSPHS (theorem), 1974
PSS (subsection, section SSLE), 1975
PSSD (theorem), 1976
PSSLS (theorem), 1977
PT (section), 1978
PTFP (example), 1979
PTM (example), 1980
PTMEE (example), 1981
PTMT (theorem), 1982
Q (archetype), 1983
R (archetype), 1984
R (chapter), 1985
range
full
example FRAN, 1986
isomorphic to column space
theorem RCSI, 1987
linear transformation
example RAO, 1988
notation, 1989
of a linear transformation
definition RLT, 1990
pre-image
theorem RPI, 1991
subspace
theorem RLTS, 1992
surjective linear transformation
theorem RSLT, 1993
via matrix representation
example RVMR, 1994
rank
computing
theorem CRN, 1995
linear transformation
definition ROLT, 1996
matrix
definition ROM, 1997
example RNM, 1998
notation, 1999, 2000
of transpose
example RRTI, 2001
square matrix
example RNSM, 2002
surjective linear transformation
theorem ROSLT, 2003
transpose
theorem RMRT, 2004
rank one decomposition
size 2
example ROD2, 2005
size 4
example ROD4, 2006
theorem ROD, 2007
rank+nullity
theorem RPNC, 2008
RAO (example), 2009
RCLS (theorem), 2010
RCSI (theorem), 2011
RD (subsection, section VS), 2012
RDS (theorem), 2013
READ (subsection, section B), 2014
READ (subsection, section CB), 2015
READ (subsection, section CRS), 2016
READ (subsection, section D), 2017
READ (subsection, section DM), 2018
READ (subsection, section EE), 2019
READ (subsection, section FS), 2020
READ (subsection, section HSE), 2021
READ (subsection, section ILT), 2022
READ (subsection, section IVLT), 2023
READ (subsection, section LC), 2024
READ (subsection, section LDS), 2025
READ (subsection, section LI), 2026
READ (subsection, section LISS), 2027
READ (subsection, section LT), 2028
READ (subsection, section MINM), 2029
READ (subsection, section MISLE), 2030
READ (subsection, section MM), 2031
READ (subsection, section MO), 2032
READ (subsection, section MR), 2033
READ (subsection, section NM), 2034
READ (subsection, section O), 2035
READ (subsection, section PD), 2036
READ (subsection, section PDM), 2037
READ (subsection, section PEE), 2038
READ (subsection, section RREF), 2039
READ (subsection, section S), 2040
READ (subsection, section SD), 2041
READ (subsection, section SLT), 2042
READ (subsection, section SS), 2043
READ (subsection, section SSLE), 2044
READ (subsection, section TSS), 2045
READ (subsection, section VO), 2046
READ (subsection, section VR), 2047
READ (subsection, section VS), 2048
READ (subsection, section WILA), 2049
reduced row-echelon form
analysis
notation, 2050
definition RREF, 2051
example NRREF, 2052
example RREF, 2053
extended
definition EEF, 2054
notation
example RREFN, 2055
unique
theorem RREFU, 2056
reducing a span
example RSC5, 2057
relation of linear dependence
definition RLD, 2058
definition RLDCV, 2059
REM (definition), 2060
REMEF (theorem), 2061
REMES (theorem), 2062
REMRS (theorem), 2063
RES (example), 2064
RGEN (theorem), 2065
RLD (definition), 2066
RLDCV (definition), 2067
RLT (definition), 2068
RLT (notation), 2069
RLT (subsection, section IS), 2070
RLT (subsection, section SLT), 2071
RLTS (theorem), 2072
RMRT (theorem), 2073
RNLT (subsection, section IVLT), 2074
RNM (example), 2075
RNM (subsection, section D), 2076
RNNM (subsection, section D), 2077
RNNM (theorem), 2078
RNSM (example), 2079
RO (definition), 2080
RO (notation), 2081
RO (subsection, section RREF), 2082
ROD (section), 2083
ROD (theorem), 2084
ROD2 (example), 2085
ROD4 (example), 2086
ROLT (definition), 2087
ROLT (notation), 2088
ROM (definition), 2089
ROM (notation), 2090
ROSLT (theorem), 2091
row operations
definition RO, 2092
elementary matrices, 2093, 2094
notation, 2095
row reduce
mathematica, 2096
ti83, 2097
ti86, 2098
row space
Archetype I
example RSAI, 2099
as column space, 2100
basis
example RSB, 2101
theorem BRS, 2102
matrix, 2103
notation, 2104
row-equivalent matrices
theorem REMRS, 2105
subspace
theorem RSMS, 2106
row-equivalent matrices
definition REM, 2107
example TREM, 2108
row space, 2109
row spaces
example RSREM, 2110
theorem REMES, 2111
row-reduce
the verb
definition RR, 2112
row-reduced matrices
theorem REMEF, 2113
RPI (theorem), 2114
RPNC (theorem), 2115
RPNDD (theorem), 2116
RR (definition), 2117
RR.MMA (computation, section MMA), 2118
RR.TI83 (computation, section TI83), 2119
RR.TI86 (computation, section TI86), 2120
RREF (definition), 2121
RREF (example), 2122
RREF (section), 2123
RREF (subsection, section RREF), 2124
RREFA (notation), 2125
RREFN (example), 2126
RREFU (theorem), 2127
RRTI (example), 2128
RS (example), 2129
RSAI (example), 2130
RSB (example), 2131
RSC5 (example), 2132
RSLT (theorem), 2133
RSM (definition), 2134
RSM (notation), 2135
RSM (subsection, section CRS), 2136
RSMS (theorem), 2137
RSNS (example), 2138
RSREM (example), 2139
RSSC4 (example), 2140
RT (subsection, section PD), 2141
RVMR (example), 2142
S (archetype), 2143
S (definition), 2144
S (example), 2145
S (section), 2146
SAA (example), 2147
SAB (example), 2148
SABMI (example), 2149
SAE (example), 2150
SAN (example), 2151
SAR (example), 2152
SAS (section), 2153
SAV (example), 2154
SC (definition), 2155
SC (example), 2156
SC (notation), 2157
SC (Property), 2158
SC (subsection, section S), 2159
SC (subsection, section SET), 2160
SC3 (example), 2161
SCAA (example), 2162
SCAB (example), 2163
SCAD (example), 2164
scalar closure
column vectors
Property SCC, 2165
matrices
Property SCM, 2166
vectors
Property SC, 2167
scalar multiple
matrix inverse, 2168
scalar multiplication
zero scalar
theorem ZSSM, 2169
zero vector
theorem ZVSM, 2170
zero vector result
theorem SMEZV, 2171
scalar multiplication associativity
column vectors
Property SMAC, 2172
matrices
Property SMAM, 2173
vectors
Property SMA, 2174
SCB (theorem), 2175
SCC (Property), 2176
SCM (Property), 2177
SD (section), 2178
SDS (example), 2179
SE (definition), 2180
SE (notation), 2181
secret sharing
6 ways
example SS6W, 2182
SEE (example), 2183
SEEF (example), 2184
SER (theorem), 2185
set
cardinality
definition C, 2186
example CS, 2187
notation, 2188
complement
definition SC, 2189
example SC, 2190
notation, 2191
definition SET, 2192
empty
definition ES, 2193
equality
definition SE, 2194
notation, 2195
intersection
definition SI, 2196
example SI, 2197
notation, 2198
membership
example SETM, 2199
notation, 2200
size, 2201
subset, 2202
union
definition SU, 2203
example SU, 2204
notation, 2205
SET (definition), 2206
SET (section), 2207
SETM (example), 2208
SETM (notation), 2209
shoes, 2210
SHS (subsection, section HSE), 2211
SI (definition), 2212
SI (example), 2213
SI (notation), 2214
SI (subsection, section IVLT), 2215
SIM (definition), 2216
similar matrices
equal eigenvalues
example EENS, 2217
eual eigenvalues
theorem SMEE, 2218
example SMS3, 2219
example SMS5, 2220
similarity
definition SIM, 2221
equivalence relation
theorem SER, 2222
singular matrix
Archetype A
example S, 2223
null space
example NSS, 2224
singular matrix, row-reduced
example SRR, 2225
singular value decomposition
theorem SVD, 2226
singular values
definition SV, 2227
SLE (chapter), 2228
SLE (definition), 2229
SLE (subsection, section SSLE), 2230
SLELT (subsection, section IVLT), 2231
SLEMM (theorem), 2232
SLSLC (theorem), 2233
SLT (definition), 2234
SLT (section), 2235
SLTB (theorem), 2236
SLTD (subsection, section SLT), 2237
SLTD (theorem), 2238
SLTLT (theorem), 2239
SM (definition), 2240
SM (notation), 2241
SM (subsection, section SD), 2242
SM2Z7 (example), 2243
SM32 (example), 2244
SMA (Property), 2245
SMAC (Property), 2246
SMAM (Property), 2247
SMEE (theorem), 2248
SMEZV (theorem), 2249
SMLT (example), 2250
SMS (theorem), 2251
SMS3 (example), 2252
SMS5 (example), 2253
SMZD (theorem), 2254
SMZE (theorem), 2255
SNCM (theorem), 2256
SO (subsection, section SET), 2257
socks, 2258
SOL (subsection, section B), 2259
SOL (subsection, section CB), 2260
SOL (subsection, section CRS), 2261
SOL (subsection, section D), 2262
SOL (subsection, section DM), 2263
SOL (subsection, section EE), 2264
SOL (subsection, section F), 2265
SOL (subsection, section FS), 2266
SOL (subsection, section HSE), 2267
SOL (subsection, section ILT), 2268
SOL (subsection, section IVLT), 2269
SOL (subsection, section LC), 2270
SOL (subsection, section LDS), 2271
SOL (subsection, section LI), 2272
SOL (subsection, section LISS), 2273
SOL (subsection, section LT), 2274
SOL (subsection, section MINM), 2275
SOL (subsection, section MISLE), 2276
SOL (subsection, section MM), 2277
SOL (subsection, section MO), 2278
SOL (subsection, section MR), 2279
SOL (subsection, section NM), 2280
SOL (subsection, section PD), 2281
SOL (subsection, section PDM), 2282
SOL (subsection, section PEE), 2283
SOL (subsection, section RREF), 2284
SOL (subsection, section S), 2285
SOL (subsection, section SD), 2286
SOL (subsection, section SLT), 2287
SOL (subsection, section SS), 2288
SOL (subsection, section SSLE), 2289
SOL (subsection, section T), 2290
SOL (subsection, section TSS), 2291
SOL (subsection, section VO), 2292
SOL (subsection, section VR), 2293
SOL (subsection, section VS), 2294
SOL (subsection, section WILA), 2295
solution set
Archetype A
example SAA, 2296
archetype E
example SAE, 2297
theorem PSPHS, 2298
solution sets
possibilities
theorem PSSLS, 2299
solution vector
definition SV, 2300
solving homogeneous system
Archetype A
example HISAA, 2301
Archetype B
example HUSAB, 2302
Archetype D
example HISAD, 2303
solving nonlinear equations
example STNE, 2304
SP4 (example), 2305
span
basic
example ABS, 2306
basis
theorem BS, 2307
definition SS, 2308
definition SSCV, 2309
improved
example IAS, 2310
notation, 2311
reducing
example RSSC4, 2312
reduction
example RS, 2313
removing vectors
example COV, 2314
reworking elements
example RES, 2315
set of polynomials
example SSP, 2316
subspace
theorem SSS, 2317
span of columns
Archetype A
example SCAA, 2318
Archetype B
example SCAB, 2319
Archetype D
example SCAD, 2320
spanning set
crazy vector space
example SSC, 2321
definition TSVS, 2322
matrices
example SSM22, 2323
more vectors
theorem SSLD, 2324
polynomials
example SSP4, 2325
SPIAS (example), 2326
SQM (definition), 2327
square root
eigenvalues, eigenspaces
theorem EESR, 2328
matrix
definition SRM, 2329
notation, 2330
positive semi-definite matrix
theorem PSMSR, 2331
unique
theorem USR, 2332
SR (section), 2333
SRM (definition), 2334
SRM (notation), 2335
SRM (subsection, section SR), 2336
SRR (example), 2337
SS (definition), 2338
SS (example), 2339
SS (section), 2340
SS (subsection, section LISS), 2341
SS (theorem), 2342
SS6W (example), 2343
SSC (example), 2344
SSCV (definition), 2345
SSET (definition), 2346
SSET (example), 2347
SSET (notation), 2348
SSLD (theorem), 2349
SSLE (section), 2350
SSM22 (example), 2351
SSNS (example), 2352
SSNS (subsection, section SS), 2353
SSNS (theorem), 2354
SSP (example), 2355
SSP4 (example), 2356
SSRLT (theorem), 2357
SSS (theorem), 2358
SSSLT (subsection, section SLT), 2359
SSV (notation), 2360
SSV (subsection, section SS), 2361
starting proofs
technique GS, 2362
STLT (example), 2363
STNE (example), 2364
SU (definition), 2365
SU (example), 2366
SU (notation), 2367
submatrix
notation, 2368
subset
definition SSET, 2369
notation, 2370
subspace
as null space
example RSNS, 2371
characterized
example ASC, 2372
definition S, 2373
in
example SP4, 2374
not, additive closure
example NSC2A, 2375
not, scalar closure
example NSC2S, 2376
not, zero vector
example NSC2Z, 2377
testing
theorem TSS, 2378
trivial
definition TS, 2379
verification
example SC3, 2380
example SM32, 2381
subspaces
equal dimension
theorem EDYES, 2382
surjective
Archetype N
example SAN, 2383
example SAR, 2384
not
example NSAQ, 2385
example NSAQR, 2386
not, Archetype O
example NSAO, 2387
not, by dimension
example NSDAT, 2388
polynomials to matrices
example SAV, 2389
surjective linear transformation
bases
theorem SLTB, 2390
surjective linear transformations
dimension
theorem SLTD, 2391
SUV (definition), 2392
SUVB (theorem), 2393
SUVOS (example), 2394
SV (definition), 2395, 2396
SVD (section), 2397
SVD (subsection, section SVD), 2398
SVD (theorem), 2399
SVP4 (example), 2400
SYM (definition), 2401
SYM (example), 2402
symmetric matrices
theorem SMS, 2403
symmetric matrix
example SYM, 2404
system of equations
vector equality
example VESE, 2405
system of linear equations
definition SLE, 2406
T (archetype), 2407
T (definition), 2408
T (notation), 2409
T (part), 2410
T (section), 2411
T (technique, section PT), 2412
TCSD (example), 2413
TD (section), 2414
TD (subsection, section TD), 2415
TD (theorem), 2416
TD4 (example), 2417
TDEE (theorem), 2418
TDEE6 (example), 2419
TDSSE (example), 2420
TDSSE (subsection, section TD), 2421
technique
C, 2422
CD, 2423
CP, 2424
CV, 2425
D, 2426
DC, 2427
E, 2428
GS, 2429
I, 2430
L, 2431
LC, 2432
ME, 2433
N, 2434
P, 2435
PI, 2436
T, 2437
U, 2438
theorem
AA, 2439
AIP, 2440
AISM, 2441
AIU, 2442
AMA, 2443
AMSM, 2444
BCS, 2445
BIS, 2446
BNS, 2447
BRS, 2448
BS, 2449
CB, 2450
CBOB, 2451
CCM, 2452
CCRA, 2453
CCRM, 2454
CCT, 2455
CFDVS, 2456
CFNLT, 2457
CHT, 2458
CILTI, 2459
CINM, 2460
CIVLT, 2461
CLI, 2462
CLTLT, 2463
CMVEI, 2464
CNMB, 2465
COB, 2466
CPSM, 2467
CRMA, 2468
CRMSM, 2469
CRN, 2470
CRSM, 2471
CRVA, 2472
CSCS, 2473
CSLTS, 2474
CSMS, 2475
CSNM, 2476
CSRN, 2477
CSRST, 2478
CSS, 2479
CUMOS, 2480
DC, 2481
DCM, 2482
DCP, 2483
DEC, 2484
DED, 2485
DEM, 2486
DEMMM, 2487
DER, 2488
DERC, 2489
DFS, 2490
DGES, 2491
DIM, 2492
DLDS, 2493
DM, 2494
DMFE, 2495
DMHP, 2496
DMMP, 2497
DMST, 2498
DNLT, 2499
DP, 2500
DRCM, 2501
DRCMA, 2502
DRCS, 2503
DRMM, 2504
DSD, 2505
DSFB, 2506
DSFOS, 2507
DSLI, 2508
DSZI, 2509
DSZV, 2510
DT, 2511
DVM, 2512
DZRC, 2513
EDELI, 2514
EDYES, 2515
EEMAP, 2516
EER, 2517
EESR, 2518
EIM, 2519
EIS, 2520
ELIS, 2521
EMDRO, 2522
EMHE, 2523
EMMVP, 2524
EMN, 2525
EMNS, 2526
EMP, 2527
EMRCP, 2528
EMS, 2529
ENLT, 2530
EOMP, 2531
EOPSS, 2532
EPM, 2533
EPSM, 2534
ERMCP, 2535
ESMM, 2536
ETM, 2537
FIMP, 2538
FS, 2539
FTMR, 2540
FVCS, 2541
G, 2542
GEK, 2543
GESD, 2544
GESIS, 2545
GSP, 2546
HMIP, 2547
HMOE, 2548
HMRE, 2549
HMVEI, 2550
HPC, 2551
HPDAA, 2552
HPHI, 2553
HPHID, 2554
HPSMM, 2555
HSC, 2556
ICBM, 2557
ICLT, 2558
IFDVS, 2559
IILT, 2560
ILTB, 2561
ILTD, 2562
ILTIS, 2563
ILTLI, 2564
ILTLT, 2565
IMILT, 2566
IMR, 2567
IP, 2568
IPAC, 2569
IPN, 2570
IPSM, 2571
IPVA, 2572
ISRN, 2573
ITMT, 2574
IVSED, 2575
JCFLT, 2576
KILT, 2577
KLTS, 2578
KNSI, 2579
KPI, 2580
KPIS, 2581
KPLT, 2582
KPNLT, 2583
LIVHS, 2584
LIVRN, 2585
LNSMS, 2586
LSMR, 2587
LTDB, 2588
LTLC, 2589
LTTZZ, 2590
MBLT, 2591
MCT, 2592
ME, 2593
MIMI, 2594
MISM, 2595
MIT, 2596
MIU, 2597
MLTCV, 2598
MLTLT, 2599
MMA, 2600
MMAD, 2601
MMCC, 2602
MMDAA, 2603
MMIM, 2604
MMIP, 2605
MMSMM, 2606
MMT, 2607
MMZM, 2608
MNEM, 2609
MRCB, 2610
MRCLT, 2611
MRMLT, 2612
MRRGE, 2613
MRSLT, 2614
MVSLD, 2615
NEM, 2616
NI, 2617
NJB, 2618
NME1, 2619
NME2, 2620
NME3, 2621
NME4, 2622
NME5, 2623
NME6, 2624
NME7, 2625
NME8, 2626
NME9, 2627
NMLIC, 2628
NMPEM, 2629
NMRRI, 2630
NMTNS, 2631
NMUS, 2632
NOILT, 2633
NPNT, 2634
NSMS, 2635
NVM, 2636
OBNM, 2637
OBUTR, 2638
OD, 2639
OSIS, 2640
OSLI, 2641
PCNA, 2642
PDM, 2643
PEEF, 2644
PIP, 2645
PSMSR, 2646
PSPHS, 2647
PSSD, 2648
PSSLS, 2649
PTMT, 2650
RCLS, 2651
RCSI, 2652
RDS, 2653
REMEF, 2654
REMES, 2655
REMRS, 2656
RGEN, 2657
RLTS, 2658
RMRT, 2659
RNNM, 2660
ROD, 2661
ROSLT, 2662
RPI, 2663
RPNC, 2664
RPNDD, 2665
RREFU, 2666
RSLT, 2667
RSMS, 2668
SCB, 2669
SER, 2670
SLEMM, 2671
SLSLC, 2672
SLTB, 2673
SLTD, 2674
SLTLT, 2675
SMEE, 2676
SMEZV, 2677
SMS, 2678
SMZD, 2679
SMZE, 2680
SNCM, 2681
SS, 2682
SSLD, 2683
SSNS, 2684
SSRLT, 2685
SSS, 2686
SUVB, 2687
SVD, 2688
TD, 2689
TDEE, 2690
technique T, 2691
TIST, 2692
TL, 2693
TMA, 2694
TMSM, 2695
TSE, 2696
TSRM, 2697
TSS, 2698
TT, 2699
TTMI, 2700
UMCOB, 2701
UMI, 2702
UMPIP, 2703
USR, 2704
UTMR, 2705
VFSLS, 2706
VRI, 2707
VRILT, 2708
VRLT, 2709
VRRB, 2710
VRS, 2711
VSLT, 2712
VSPCV, 2713
VSPM, 2714
ZSSM, 2715
ZVSM, 2716
ZVU, 2717
ti83
matrix entry (computation), 2718
row reduce (computation), 2719
vector linear combinations (computation), 2720
TI83 (section), 2721
ti86
matrix entry (computation), 2722
row reduce (computation), 2723
transpose of a matrix (computation), 2724
vector linear combinations (computation), 2725
TI86 (section), 2726
TIS (example), 2727
TIST (theorem), 2728
TIVS (example), 2729
TKAP (example), 2730
TL (theorem), 2731
TLC (example), 2732
TM (definition), 2733
TM (example), 2734
TM (notation), 2735
TM (subsection, section OD), 2736
TM.MMA (computation, section MMA), 2737
TM.TI86 (computation, section TI86), 2738
TMA (theorem), 2739
TMP (example), 2740
TMSM (theorem), 2741
TOV (example), 2742
trace
definition T, 2743
linearity
theorem TL, 2744
matrix multiplication
theorem TSRM, 2745
notation, 2746
similarity
theorem TIST, 2747
sum of eigenvalues
theorem TSE, 2748
trail mix
example TMP, 2749
transpose
matrix scalar multiplication
theorem TMSM, 2750
example TM, 2751
matrix addition
theorem TMA, 2752
matrix inverse, 2753, 2754
notation, 2755
scalar multiplication, 2756
transpose of a matrix
mathematica, 2757
ti86, 2758
transpose of a transpose
theorem TT, 2759
TREM (example), 2760
triangular decomposition
entry by entry, size 6
example TDEE6, 2761
entry by entry
theorem TDEE, 2762
size 4
example TD4, 2763
solving systems of equations
example TDSSE, 2764
theorem TD, 2765
triangular matrix
inverse
theorem ITMT, 2766
trivial solution
system of equations
definition TSHSE, 2767
TS (definition), 2768
TS (subsection, section S), 2769
TSE (theorem), 2770
TSHSE (definition), 2771
TSM (subsection, section MO), 2772
TSRM (theorem), 2773
TSS (section), 2774
TSS (subsection, section S), 2775
TSS (theorem), 2776
TSVS (definition), 2777
TT (theorem), 2778
TTMI (theorem), 2779
TTS (example), 2780
typical systems,
example TTS, 2781
U (archetype), 2782
U (technique, section PT), 2783
UM (definition), 2784
UM (subsection, section MINM), 2785
UM3 (example), 2786
UMCOB (theorem), 2787
UMI (theorem), 2788
UMPIP (theorem), 2789
unique solution,
example US, 2790
example USR, 2791
uniqueness
technique U, 2792
unit vectors
basis
theorem SUVB, 2793
definition SUV, 2794
orthogonal
example SUVOS, 2795
unitary
permutation matrix
example UPM, 2796
size 3
example UM3, 2797
unitary matrices
columns
theorem CUMOS, 2798
unitary matrix
inner product
theorem UMPIP, 2799
UPM (example), 2800
upper triangular matrix
definition UTM, 2801
URREF (subsection, section LC), 2802
US (example), 2803
USR (example), 2804
USR (theorem), 2805
UTM (definition), 2806
UTMR (subsection, section OD), 2807
UTMR (theorem), 2808
V (archetype), 2809
V (chapter), 2810
VA (example), 2811
Vandermonde matrix
definition VM, 2812
vandermonde matrix
determinant
theorem DVM, 2813
nonsingular
theorem NVM, 2814
size 4
example VM4, 2815
VEASM (subsection, section VO), 2816
vector
addition
definition CVA, 2817
column
definition CV, 2818
equality
definition CVE, 2819
notation, 2820
inner product
definition IP, 2821
norm
definition NV, 2822
notation, 2823
of constants
definition VOC, 2824
product with matrix, 2825, 2826
scalar multiplication
definition CVSM, 2827
vector addition
example VA, 2828
vector component
notation, 2829
vector form of solutions
Archetype D
example VFSAD, 2830
Archetype I
example VFSAI, 2831
Archetype L
example VFSAL, 2832
example VFS, 2833
mathematica, 2834
theorem VFSLS, 2835
vector linear combinations
mathematica, 2836
ti83, 2837
ti86, 2838
vector representation
example AVR, 2839
example VRC4, 2840
injective
theorem VRI, 2841
invertible
theorem VRILT, 2842
linear transformation
definition VR, 2843
theorem VRLT, 2844
surjective
theorem VRS, 2845
theorem VRRB, 2846
vector representations
polynomials
example VRP2, 2847
vector scalar multiplication
example CVSM, 2848
vector space
characterization
theorem CFDVS, 2849
column vectors
definition VSCV, 2850
definition VS, 2851
infinite dimension
example VSPUD, 2852
linear transformations
theorem VSLT, 2853
over integers mod 5
example VSIM5, 2854
vector space of column vectors
notation, 2855
vector space of functions
example VSF, 2856
vector space of infinite sequences
example VSIS, 2857
vector space of matrices
definition VSM, 2858
example VSM, 2859
notation, 2860
vector space of polynomials
example VSP, 2861
vector space properties
column vectors
theorem VSPCV, 2862
matrices
theorem VSPM, 2863
vector space, crazy
example CVS, 2864
vector space, singleton
example VSS, 2865
vector spaces
isomorphic
definition IVS, 2866
theorem IFDVS, 2867
VESE (example), 2868
VFS (example), 2869
VFSAD (example), 2870
VFSAI (example), 2871
VFSAL (example), 2872
VFSLS (theorem), 2873
VFSS (subsection, section LC), 2874
VFSS.MMA (computation, section MMA), 2875
VLC.MMA (computation, section MMA), 2876
VLC.TI83 (computation, section TI83), 2877
VLC.TI86 (computation, section TI86), 2878
VM (definition), 2879
VM (section), 2880
VM4 (example), 2881
VO (section), 2882
VOC (definition), 2883
VR (definition), 2884
VR (section), 2885
VR (subsection, section LISS), 2886
VRC4 (example), 2887
VRI (theorem), 2888
VRILT (theorem), 2889
VRLT (theorem), 2890
VRP2 (example), 2891
VRRB (theorem), 2892
VRS (theorem), 2893
VS (chapter), 2894
VS (definition), 2895
VS (section), 2896
VS (subsection, section VS), 2897
VSCV (definition), 2898
VSCV (example), 2899
VSCV (notation), 2900
VSF (example), 2901
VSIM5 (example), 2902
VSIS (example), 2903
VSLT (theorem), 2904
VSM (definition), 2905
VSM (example), 2906
VSM (notation), 2907
VSP (example), 2908
VSP (subsection, section MO), 2909
VSP (subsection, section VO), 2910
VSP (subsection, section VS), 2911
VSPCV (theorem), 2912
VSPM (theorem), 2913
VSPUD (example), 2914
VSS (example), 2915
W (archetype), 2916
WILA (section), 2917
X (archetype), 2918
Z (Property), 2919
ZC (Property), 2920
ZCN (Property), 2921
ZCV (definition), 2922
ZCV (notation), 2923
zero
complex numbers
Property ZCN, 2924
field
Property ZF, 2925
zero column vector
definition ZCV, 2926
notation, 2927
zero matrix
notation, 2928
zero vector
column vectors
Property ZC, 2929
matrices
Property ZM, 2930
unique
theorem ZVU, 2931
vectors
Property Z, 2932
ZF (Property), 2933
ZM (definition), 2934
ZM (notation), 2935
ZM (Property), 2936
ZNDAB (example), 2937
ZSSM (theorem), 2938
ZVSM (theorem), 2939
ZVU (theorem), 2940