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