B (archetype), 80
B (definition), 81
B (section), 82
B (subsection, section B), 83
basis
columns nonsingular matrix
example CABAK, 84
common size
theorem BIS, 85
crazy vector apace
example BC, 86
definition B, 87
extending
theorem EB, 88
matrices
example BM, 89
example BSM22, 90
polynomials
example BP, 91
example BPR, 92
example BSP4, 93
example SVP4, 94
subspace of matrices
example BDM22, 95
BC (example), 96
BCS (theorem), 97
BDE (example), 98
BDM22 (example), 99
best cities
money magazine
example MBC, 100
BIS (theorem), 101
BM (example), 102
BNM (subsection, section B), 103
BNS (theorem), 104
BP (example), 105
BPR (example), 106
BRLT (example), 107
BRS (theorem), 108
BS (theorem), 109
BSCV (subsection, section B), 110
BSM22 (example), 111
BSP4 (example), 112
C (archetype), 113
C (definition), 114
C (notation), 115
C (part), 116
C (Property), 117
C (technique, section PT), 118
CABAK (example), 119
CAEHW (example), 120
cancellation
vector addition
theorem VAC, 121
CAV (subsection, section O), 122
CB (section), 123
CB (theorem), 124
CBCV (example), 125
CBM (definition), 126
CBM (subsection, section CB), 127
CBP (example), 128
CC (Property), 129
CCCV (definition), 130
CCCV (notation), 131
CCM (definition), 132
CCM (example), 133
CCM (notation), 134
CCN (definition), 135
CCN (notation), 136
CCN (subsection, section CNO), 137
CCRA (theorem), 138
CCRM (theorem), 139
CCT (theorem), 140
CD (subsection, section DM), 141
CD (technique, section PT), 142
CEE (subsection, section EE), 143
CELT (example), 144
CELT (subsection, section CB), 145
CEMS6 (example), 146
CFDVS (theorem), 147
CFV (example), 148
change of basis
between polynomials
example CBP, 149
change-of-basis
between column vectors
example CBCV, 150
matrix representation
theorem MRCB, 151
similarity
theorem SCB, 152
theorem CB, 153
change-of-basis matrix
definition CBM, 154
inverse
theorem ICBM, 155
characteristic polynomial
definition CP, 156
degree
theorem DCP, 157
size 3 matrix
example CPMS3, 158
CILT (subsection, section ILT), 159
CILTI (theorem), 160
CIM (subsection, section MISLE), 161
CINM (theorem), 162
CIVLT (theorem), 163
CLI (theorem), 164
CLTLT (theorem), 165
CM (definition), 166
CM (Property), 167
CM32 (example), 168
CMI (example), 169
CMIAB (example), 170
CMVEI (theorem), 171
CN (appendix), 172
CNA (definition), 173
CNA (notation), 174
CNA (subsection, section CNO), 175
CNE (definition), 176
CNE (notation), 177
CNM (definition), 178
CNM (notation), 179
CNMB (theorem), 180
CNO (section), 181
CNS1 (example), 182
CNS2 (example), 183
CNSV (example), 184
COB (theorem), 185
coefficient matrix
definition CM, 186
nonsingular
theorem SNCM, 187
column space
as null space
theorem FS, 188
Archetype A
example CSAA, 189
Archetype B
example CSAB, 190
as null space
example CSANS, 191
as null space, Archetype G
example FSAG, 192
as row space
theorem CSRST, 193
basis
theorem BCS, 194
consistent system
theorem CSCS, 195
consistent systems
example CSMCS, 196
isomorphic to range, 197
matrix, 198
nonsingular matrix
theorem CSNM, 199
notation, 200
original columns, Archetype D
example CSOCD, 201
row operations, Archetype I
example CSROI, 202
subspace
theorem CSMS, 203
testing membership
example MCSM, 204
two computations
example CSTW, 205
column vector addition
notation, 206
column vector scalar multiplication
notation, 207
commutativity
column vectors
Property CC, 208
matrices
Property CM, 209
vectors
Property C, 210
complex -space
example VSCV, 211
complex arithmetic
example ACN, 212
complex number
conjugate
example CSCN, 213
modulus
example MSCN, 214
complex number
conjugate
definition CCN, 215
modulus
definition MCN, 216
complex numbers
addition
definition CNA, 217
notation, 218
arithmetic properties
theorem PCNA, 219
equality
definition CNE, 220
notation, 221
multiplication
definition CNM, 222
notation, 223
complex vector space
dimension
theorem DCM, 224
composition
injective linear transformations
theorem CILTI, 225
surjective linear transformations
theorem CSLTS, 226
conjugate
addition
theorem CCRA, 227
column vector
definition CCCV, 228
matrix
definition CCM, 229
notation, 230
multiplication
theorem CCRM, 231
notation, 232
scalar multiplication
theorem CRSM, 233
twice
theorem CCT, 234
vector addition
theorem CRVA, 235
conjugate of a vector
notation, 236
conjugation
matrix addition
theorem CRMA, 237
matrix scalar multiplication
theorem CRMSM, 238
matrix transpose
theorem MCT, 239
consistent linear system, 240
consistent linear systems
theorem CSRN, 241
consistent system
definition CS, 242
constructive proofs
technique C, 243
contradiction
technique CD, 244
contrapositive
technique CP, 245
converse
technique CV, 246
coordinates
orthonormal basis
theorem COB, 247
coordinatization
linear combination of matrices
example CM32, 248
linear independence
theorem CLI, 249
orthonormal basis
example CROB3, 250
example CROB4, 251
spanning sets
theorem CSS, 252
coordinatization principle, 253
coordinatizing
polynomials
example CP2, 254
COV (example), 255
COV (subsection, section LDS), 256
CP (definition), 257
CP (subsection, section VR), 258
CP (technique, section PT), 259
CP2 (example), 260
CPMS3 (example), 261
crazy vector space
example CVSR, 262
properties
example PCVS, 263
CRMA (theorem), 264
CRMSM (theorem), 265
CRN (theorem), 266
CROB3 (example), 267
CROB4 (example), 268
CRS (section), 269
CRS (subsection, section FS), 270
CRSM (theorem), 271
CRVA (theorem), 272
CS (definition), 273
CS (example), 274
CS (subsection, section TSS), 275
CSAA (example), 276
CSAB (example), 277
CSANS (example), 278
CSCN (example), 279
CSCS (theorem), 280
CSIP (example), 281
CSLT (subsection, section SLT), 282
CSLTS (theorem), 283
CSM (definition), 284
CSM (notation), 285
CSMCS (example), 286
CSMS (theorem), 287
CSNM (subsection, section CRS), 288
CSNM (theorem), 289
CSOCD (example), 290
CSRN (theorem), 291
CSROI (example), 292
CSRST (theorem), 293
CSS (theorem), 294
CSSE (subsection, section CRS), 295
CSSM (theorem), 296
CSSOC (subsection, section CRS), 297
CSTW (example), 298
CTLT (example), 299
CUMOS (theorem), 300
CV (definition), 301
CV (notation), 302
CV (technique, section PT), 303
CVA (definition), 304
CVA (notation), 305
CVC (notation), 306
CVE (definition), 307
CVE (notation), 308
CVS (example), 309
CVS (subsection, section VR), 310
CVSM (definition), 311
CVSM (example), 312
CVSM (notation), 313
CVSM (theorem), 314
CVSR (example), 315
D (archetype), 316
D (chapter), 317
D (definition), 318
D (notation), 319
D (section), 320
D (subsection, section D), 321
D (subsection, section SD), 322
D (technique, section PT), 323
D33M (example), 324
DAB (example), 325
DC (example), 326
DC (technique, section PT), 327
DC (theorem), 328
DCM (theorem), 329
DCN (Property), 330
DCP (theorem), 331
DD (subsection, section DM), 332
DEC (theorem), 333
decomposition
technique DC, 334
DED (theorem), 335
definition
A, 336
AM, 337
AME, 338
B, 339
C, 340
CBM, 341
CCCV, 342
CCM, 343
CCN, 344
CM, 345
CNA, 346
CNE, 347
CNM, 348
CP, 349
CS, 350
CSM, 351
CV, 352
CVA, 353
CVE, 354
CVSM, 355
D, 356
DIM, 357
DM, 358
DS, 359
DZM, 360
EEF, 361
EELT, 362
EEM, 363
ELEM, 364
EM, 365
EO, 366
ES, 367
ESYS, 368
GME, 369
HM, 370
HS, 371
IDLT, 372
IDV, 373
ILT, 374
IM, 375
IP, 376
IVLT, 377
IVS, 378
KLT, 379
LC, 380
LCCV, 381
LI, 382
LICV, 383
LNS, 384
LSMR, 385
LT, 386
LTA, 387
LTC, 388
LTSM, 389
M, 390
MA, 391
MCN, 392
ME, 393
MI, 394
MM, 395
MR, 396
MSM, 397
MVP, 398
NM, 399
NOLT, 400
NOM, 401
NSM, 402
NV, 403
ONS, 404
OSV, 405
OV, 406
PI, 407
REM, 408
RLD, 409
RLDCV, 410
RLT, 411
RO, 412
ROLT, 413
ROM, 414
RR, 415
RREF, 416
RSM, 417
S, 418
SC, 419
SE, 420
SET, 421
SI, 422
SIM, 423
SLE, 424
SLT, 425
SM, 426
SQM, 427
SS, 428
SSCV, 429
SSET, 430
SU, 431
SUV, 432
SV, 433
SYM, 434
technique D, 435
TM, 436
TS, 437
TSHSE, 438
TSVS, 439
UM, 440
VOC, 441
VR, 442
VS, 443
VSCV, 444
VSM, 445
ZCV, 446
ZM, 447
DEHD (example), 448
DEM (theorem), 449
DEMMM (theorem), 450
DEMS5 (example), 451
DER (theorem), 452
DERC (theorem), 453
determinant
computed two ways
example TCSD, 454
definition DM, 455
equal rows or columns
theorem DERC, 456
expansion, columns
theorem DEC, 457
expansion, rows
theorem DER, 458
identity matrix
theorem DIM, 459
matrix multiplication
theorem DRMM, 460
nonsingular matrix, 461
notation, 462
row or column multiple
theorem DRCM, 463
row or column swap
theorem DRCS, 464
size 2 matrix
theorem DMST, 465
size 3 matrix
example D33M, 466
transpose
theorem DT, 467
via row operations
example DRO, 468
zero
theorem SMZD, 469
zero row or column
theorem DZRC, 470
zero versus nonzero
example ZNDAB, 471
determinant, upper triangular matrix
example DUTM, 472
determinants
elementary matrices
theorem DEMMM, 473
DFS (subsection, section PD), 474
DFS (theorem), 475
diagonal matrix
definition DIM, 476
diagonalizable
definition DZM, 477
distinct eigenvalues
example DEHD, 478
theorem DED, 479
full eigenspaces
theorem DMFE, 480
not
example NDMS4, 481
diagonalizable matrix
high power
example HPDM, 482
diagonalization
Archetype B
example DAB, 483
criteria
theorem DC, 484
example DMS3, 485
DIM (definition), 486
DIM (theorem), 487
dimension
crazy vector space
example DC, 488
definition D, 489
notation, 490
polynomial subspace
example DSP4, 491
proper subspaces
theorem PSSD, 492
subspace
example DSM22, 493
direct sum
definition DS, 494
notation, 495
distributivity
complex numbers
Property DCN, 496
distributivity, matrix addition
matrices
Property DMAM, 497
distributivity, scalar addition
column vectors
Property DSAC, 498
matrices
Property DSAM, 499
vectors
Property DSA, 500
distributivity, vector addition
column vectors
Property DVAC, 501
vectors
Property DVA, 502
DLDS (theorem), 503
DM (definition), 504
DM (notation), 505
DM (section), 506
DM (theorem), 507
DMAM (Property), 508
DMFE (theorem), 509
DMS3 (example), 510
DMST (theorem), 511
DNMMM (subsection, section PDM), 512
DP (theorem), 513
DRCM (theorem), 514
DRCMA (theorem), 515
DRCS (theorem), 516
DRMM (theorem), 517
DRO (example), 518
DRO (subsection, section PDM), 519
DROEM (subsection, section PDM), 520
DS (definition), 521
DS (notation), 522
DSA (Property), 523
DSAC (Property), 524
DSAM (Property), 525
DSM22 (example), 526
DSP4 (example), 527
DT (theorem), 528
DUTM (example), 529
DVA (Property), 530
DVAC (Property), 531
DVS (subsection, section D), 532
DZM (definition), 533
DZRC (theorem), 534
E (archetype), 535
E (chapter), 536
E (technique, section PT), 537
EB (theorem), 538
ECEE (subsection, section EE), 539
EDELI (theorem), 540
EDYES (theorem), 541
EE (section), 542
EEE (subsection, section EE), 543
EEF (definition), 544
EEF (subsection, section FS), 545
EELT (definition), 546
EELT (subsection, section CB), 547
EEM (definition), 548
EEM (subsection, section EE), 549
EENS (example), 550
EER (theorem), 551
EHM (subsection, section PEE), 552
eigenspace
as null space
theorem EMNS, 553
definition EM, 554
subspace
theorem EMS, 555
eigenvalue
algebraic multiplicity
definition AME, 556
complex
example CEMS6, 557
definition EEM, 558
existence
example CAEHW, 559
theorem EMHE, 560
geometric multiplicity
definition GME, 561
linear transformation
definition EELT, 562
multiplicities
example EMMS4, 563
power
theorem EOMP, 564
root of characteristic polynomial
theorem EMRCP, 565
scalar multiple
theorem ESMM, 566
symmetric matrix
example ESMS4, 567
zero
theorem SMZE, 568
eigenvalues
building desired
example BDE, 569
complex, of a linear transformation
example CELT, 570
conjugate pairs
theorem ERMCP, 571
distinct
example DEMS5, 572
example SEE, 573
Hermitian matrices
theorem HMRE, 574
inverse
theorem EIM, 575
maximum number
theorem MNEM, 576
multiplicities
example HMEM5, 577
theorem ME, 578
number
theorem NEM, 579
of a polynomial
theorem EPM, 580
size 3 matrix
example EMS3, 581
example ESMS3, 582
transpose
theorem ETM, 583
eigenvalues, eigenvectors
vector, matrix representations
theorem EER, 584
eigenvector, 585
linear transformation, 586
eigenvectors, 587
conjugate pairs, 588
Hermitian matrices
theorem HMOE, 589
linear transformation
example ELTBM, 590
example ELTBP, 591
linearly independent
theorem EDELI, 592
of a linear transformation
example ELTT, 593
EILT (subsection, section ILT), 594
EIM (theorem), 595
ELEM (definition), 596
ELEM (notation), 597
elementary matrices
definition ELEM, 598
determinants
theorem DEM, 599
nonsingular
theorem EMN, 600
notation, 601
row operations
example EMRO, 602
theorem EMDRO, 603
ELIS (theorem), 604
ELTBM (example), 605
ELTBP (example), 606
ELTT (example), 607
EM (definition), 608
EM (subsection, section DM), 609
EMDRO (theorem), 610
EMHE (theorem), 611
EMMS4 (example), 612
EMMVP (theorem), 613
EMN (theorem), 614
EMNS (theorem), 615
EMP (theorem), 616
empty set, 617
notation, 618
EMRCP (theorem), 619
EMRO (example), 620
EMS (theorem), 621
EMS3 (example), 622
EO (definition), 623
EOMP (theorem), 624
EOPSS (theorem), 625
EPM (theorem), 626
equal matrices
via equal matrix-vector products
theorem EMMVP, 627
equation operations
definition EO, 628
theorem EOPSS, 629
equivalence statements
technique E, 630
equivalences
technique ME, 631
equivalent systems
definition ESYS, 632
ERMCP (theorem), 633
ES (definition), 634
ES (notation), 635
ESEO (subsection, section SSLE), 636
ESLT (subsection, section SLT), 637
ESMM (theorem), 638
ESMS3 (example), 639
ESMS4 (example), 640
ESYS (definition), 641
ETM (theorem), 642
EVS (subsection, section VS), 643
example
AALC, 644
ABLC, 645
ABS, 646
ACN, 647
AHSAC, 648
AIVLT, 649
ALT, 650
ALTMM, 651
AM, 652
AMAA, 653
ANILT, 654
AOS, 655
ASC, 656
AVR, 657
BC, 658
BDE, 659
BDM22, 660
BM, 661
BP, 662
BPR, 663
BRLT, 664
BSM22, 665
BSP4, 666
CABAK, 667
CAEHW, 668
CBCV, 669
CBP, 670
CCM, 671
CELT, 672
CEMS6, 673
CFV, 674
CM32, 675
CMI, 676
CMIAB, 677
CNS1, 678
CNS2, 679
CNSV, 680
COV, 681
CP2, 682
CPMS3, 683
CROB3, 684
CROB4, 685
CS, 686
CSAA, 687
CSAB, 688
CSANS, 689
CSCN, 690
CSIP, 691
CSMCS, 692
CSOCD, 693
CSROI, 694
CSTW, 695
CTLT, 696
CVS, 697
CVSM, 698
CVSR, 699
D33M, 700
DAB, 701
DC, 702
DEHD, 703
DEMS5, 704
DMS3, 705
DRO, 706
DSM22, 707
DSP4, 708
DUTM, 709
EENS, 710
ELTBM, 711
ELTBP, 712
ELTT, 713
EMMS4, 714
EMRO, 715
EMS3, 716
ESMS3, 717
ESMS4, 718
FDV, 719
FRAN, 720
FS1, 721
FS2, 722
FSAG, 723
GSTV, 724
HISAA, 725
HISAD, 726
HMEM5, 727
HPDM, 728
HUSAB, 729
IAP, 730
IAR, 731
IAS, 732
IAV, 733
ILTVR, 734
IM, 735
IS, 736
ISSI, 737
IVSAV, 738
KVMR, 739
LCM, 740
LDCAA, 741
LDHS, 742
LDP4, 743
LDRN, 744
LDS, 745
LIC, 746
LICAB, 747
LIHS, 748
LIM32, 749
LINSB, 750
LIP4, 751
LIS, 752
LLDS, 753
LNS, 754
LTDB1, 755
LTDB2, 756
LTDB3, 757
LTM, 758
LTPM, 759
LTPP, 760
MA, 761
MBC, 762
MCSM, 763
MFLT, 764
MI, 765
MIVS, 766
MMNC, 767
MNSLE, 768
MOLT, 769
MPMR, 770
MRBE, 771
MRCM, 772
MSCN, 773
MSM, 774
MTV, 775
MWIAA, 776
NDMS4, 777
NIAO, 778
NIAQ, 779
NIAQR, 780
NIDAU, 781
NKAO, 782
NLT, 783
NM, 784
NRREF, 785
NSAO, 786
NSAQ, 787
NSAQR, 788
NSC2A, 789
NSC2S, 790
NSC2Z, 791
NSDAT, 792
NSDS, 793
NSE, 794
NSEAI, 795
NSLE, 796
NSLIL, 797
NSNM, 798
NSR, 799
NSS, 800
OLTTR, 801
ONFV, 802
ONTV, 803
OSGMD, 804
OSMC, 805
PCVS, 806
PM, 807
PSHS, 808
PTM, 809
PTMEE, 810
RAO, 811
RES, 812
RNM, 813
RNSM, 814
RREF, 815
RREFN, 816
RRTI, 817
RS, 818
RSAI, 819
RSB, 820
RSC5, 821
RSNS, 822
RSREM, 823
RSSC4, 824
RVMR, 825
S, 826
SAA, 827
SAB, 828
SABMI, 829
SAE, 830
SAN, 831
SAR, 832
SAV, 833
SC, 834
SC3, 835
SCAA, 836
SCAB, 837
SCAD, 838
SEE, 839
SEEF, 840
SETM, 841
SI, 842
SM32, 843
SMLT, 844
SMS3, 845
SMS5, 846
SP4, 847
SPIAS, 848
SRR, 849
SS, 850
SSC, 851
SSET, 852
SSM22, 853
SSNS, 854
SSP, 855
SSP4, 856
STLT, 857
STNE, 858
SU, 859
SUVOS, 860
SVP4, 861
SYM, 862
TCSD, 863
TIVS, 864
TKAP, 865
TLC, 866
TM, 867
TMP, 868
TOV, 869
TREM, 870
TTS, 871
UM3, 872
UPM, 873
US, 874
USR, 875
VA, 876
VESE, 877
VFS, 878
VFSAD, 879
VFSAI, 880
VFSAL, 881
VRC4, 882
VRP2, 883
VSCV, 884
VSF, 885
VSIS, 886
VSM, 887
VSP, 888
VSPUD, 889
VSS, 890
ZNDAB, 891
EXC (subsection, section B), 892
EXC (subsection, section CB), 893
EXC (subsection, section CRS), 894
EXC (subsection, section D), 895
EXC (subsection, section DM), 896
EXC (subsection, section EE), 897
EXC (subsection, section FS), 898
EXC (subsection, section HSE), 899
EXC (subsection, section ILT), 900
EXC (subsection, section IVLT), 901
EXC (subsection, section LC), 902
EXC (subsection, section LDS), 903
EXC (subsection, section LI), 904
EXC (subsection, section LISS), 905
EXC (subsection, section LT), 906
EXC (subsection, section MINM), 907
EXC (subsection, section MISLE), 908
EXC (subsection, section MM), 909
EXC (subsection, section MO), 910
EXC (subsection, section MR), 911
EXC (subsection, section NM), 912
EXC (subsection, section O), 913
EXC (subsection, section PD), 914
EXC (subsection, section PDM), 915
EXC (subsection, section PEE), 916
EXC (subsection, section RREF), 917
EXC (subsection, section S), 918
EXC (subsection, section SD), 919
EXC (subsection, section SLT), 920
EXC (subsection, section SS), 921
EXC (subsection, section SSLE), 922
EXC (subsection, section TSS), 923
EXC (subsection, section VO), 924
EXC (subsection, section VR), 925
EXC (subsection, section VS), 926
EXC (subsection, section WILA), 927
extended echelon form
submatrices
example SEEF, 928
extended reduced row-echelon form
properties
theorem PEEF, 929
F (archetype), 930
FDV (example), 931
four subsets
example FS1, 932
example FS2, 933
four subspaces
dimension
theorem DFS, 934
FRAN (example), 935
free variables
example CFV, 936
free variables, number
theorem FVCS, 937
free, independent variables
example FDV, 938
FS (section), 939
FS (subsection, section FS), 940
FS (theorem), 941
FS1 (example), 942
FS2 (example), 943
FSAG (example), 944
FTMR (theorem), 945
FV (subsection, section TSS), 946
FVCS (theorem), 947
G (archetype), 948
G (theorem), 949
GFDL (appendix), 950
GME (definition), 951
goldilocks
theorem G, 952
Gram-Schmidt
column vectors
theorem GSPCV, 953
three vectors
example GSTV, 954
gram-schmidt
mathematica, 955
GS (technique, section PT), 956
GSP (subsection, section O), 957
GSP.MMA (computation, section MMA), 958
GSPCV (theorem), 959
GSTV (example), 960
GT (subsection, section PD), 961
H (archetype), 962
hermitian
definition HM, 963
HISAA (example), 964
HISAD (example), 965
HM (definition), 966
HMEM5 (example), 967
HMOE (theorem), 968
HMRE (theorem), 969
HMVEI (theorem), 970
homogeneous system
consistent
theorem HSC, 971
definition HS, 972
infinitely many solutions
theorem HMVEI, 973
homogeneous systems
linear independence, 974
homogenous system
Archetype C
example AHSAC, 975
HPDM (example), 976
HS (definition), 977
HSC (theorem), 978
HSE (section), 979
HUSAB (example), 980
I (archetype), 981
I (technique, section PT), 982
IAP (example), 983
IAR (example), 984
IAS (example), 985
IAV (example), 986
ICBM (theorem), 987
ICLT (theorem), 988
identities
technique PI, 989
identity matrix
determinant, 990
example IM, 991
notation, 992
IDLT (definition), 993
IDV (definition), 994
IFDVS (theorem), 995
IILT (theorem), 996
ILT (definition), 997
ILT (section), 998
ILTB (theorem), 999
ILTD (subsection, section ILT), 1000
ILTD (theorem), 1001
ILTIS (theorem), 1002
ILTLI (subsection, section ILT), 1003
ILTLI (theorem), 1004
ILTLT (theorem), 1005
ILTVR (example), 1006
IM (definition), 1007
IM (example), 1008
IM (notation), 1009
IM (subsection, section MISLE), 1010
IMILT (theorem), 1011
IMR (theorem), 1012
inconsistent linear systems
theorem ISRN, 1013
independent, dependent variables
definition IDV, 1014
indesxstring
example SSET, 1015
indexstring
theorem DRCMA, 1016
induction
technique I, 1017
infinite solution set
example ISSI, 1018
infinite solutions,
example IS, 1019
injective
example IAP, 1020
example IAR, 1021
not
example NIAO, 1022
example NIAQ, 1023
example NIAQR, 1024
not, by dimension
example NIDAU, 1025
polynomials to matrices
example IAV, 1026
injective linear transformation
bases
theorem ILTB, 1027
injective linear transformations
dimension
theorem ILTD, 1028
inner product
anti-commutative
theorem IPAC, 1029
example CSIP, 1030
norm
theorem IPN, 1031
notation, 1032
positive
theorem PIP, 1033
scalar multiplication
theorem IPSM, 1034
vector addition
theorem IPVA, 1035
inverse
composition of linear transformations
theorem ICLT, 1036
example CMI, 1037
example MI, 1038
notation, 1039
of a matrix, 1040
invertible linear transformation
defined by invertible matrix
theorem IMILT, 1041
invertible linear transformations
composition
theorem CIVLT, 1042
IP (definition), 1043
IP (notation), 1044
IP (subsection, section O), 1045
IPAC (theorem), 1046
IPN (theorem), 1047
IPSM (theorem), 1048
IPVA (theorem), 1049
IS (example), 1050
isomorphic
multiple vector spaces
example MIVS, 1051
vector spaces
example IVSAV, 1052
isomorphic vector spaces
dimension
theorem IVSED, 1053
example TIVS, 1054
ISRN (theorem), 1055
ISSI (example), 1056
IV (subsection, section IVLT), 1057
IVLT (definition), 1058
IVLT (section), 1059
IVLT (subsection, section IVLT), 1060
IVLT (subsection, section MR), 1061
IVS (definition), 1062
IVSAV (example), 1063
IVSED (theorem), 1064
J (archetype), 1065
K (archetype), 1066
kernel
injective linear transformation
theorem KILT, 1067
isomorphic to null space
theorem KNSI, 1068
linear transformation
example NKAO, 1069
notation, 1070
of a linear transformation
definition KLT, 1071
pre-image, 1072
subspace
theorem KLTS, 1073
trivial
example TKAP, 1074
via matrix representation
example KVMR, 1075
KILT (theorem), 1076
KLT (definition), 1077
KLT (notation), 1078
KLT (subsection, section ILT), 1079
KLTS (theorem), 1080
KNSI (theorem), 1081
KPI (theorem), 1082
KVMR (example), 1083
L (archetype), 1084
L (technique, section PT), 1085
LA (subsection, section WILA), 1086
LC (definition), 1087
LC (section), 1088
LC (subsection, section LC), 1089
LC (technique, section PT), 1090
LCCV (definition), 1091
LCM (example), 1092
LDCAA (example), 1093
LDHS (example), 1094
LDP4 (example), 1095
LDRN (example), 1096
LDS (example), 1097
LDS (section), 1098
LDSS (subsection, section LDS), 1099
left null space
as row space, 1100
definition LNS, 1101
example LNS, 1102
notation, 1103
subspace
theorem LNSMS, 1104
lemma
technique LC, 1105
LI (definition), 1106
LI (section), 1107
LI (subsection, section LISS), 1108
LIC (example), 1109
LICAB (example), 1110
LICV (definition), 1111
LIHS (example), 1112
LIM32 (example), 1113
linear combination
system of equations
example ABLC, 1114
definition LC, 1115
definition LCCV, 1116
example TLC, 1117
linear transformation, 1118
matrices
example LCM, 1119
system of equations
example AALC, 1120
linear combinations
solutions to linear systems
theorem SLSLC, 1121
linear dependence
more vectors than size
theorem MVSLD, 1122
linear independence
definition LI, 1123
definition LICV, 1124
homogeneous systems
theorem LIVHS, 1125
injective linear transformation
theorem ILTLI, 1126
matrices
example LIM32, 1127
orthogonal, 1128
r and n
theorem LIVRN, 1129
linear solve
mathematica, 1130
linear system
consistent
theorem RCLS, 1131
matrix representation
definition LSMR, 1132
notation, 1133
linear systems
notation
example MNSLE, 1134
example NSLE, 1135
linear transformation
polynomials to polynomials
example LTPP, 1136
addition
definition LTA, 1137
theorem MLTLT, 1138
theorem SLTLT, 1139
as matrix multiplication
example ALTMM, 1140
basis of range
example BRLT, 1141
checking
example ALT, 1142
composition
definition LTC, 1143
theorem CLTLT, 1144
defined by a matrix
example LTM, 1145
defined on a basis
example LTDB1, 1146
example LTDB2, 1147
example LTDB3, 1148
theorem LTDB, 1149
definition LT, 1150
identity
definition IDLT, 1151
injection
definition ILT, 1152
inverse
theorem ILTLT, 1153
inverse of inverse
theorem IILT, 1154
invertible
definition IVLT, 1155
example AIVLT, 1156
invertible, injective and surjective
theorem ILTIS, 1157
linear combination
theorem LTLC, 1158
matrix of, 1159
example MFLT, 1160
example MOLT, 1161
not
example NLT, 1162
not invertible
example ANILT, 1163
notation, 1164
polynomials to matrices
example LTPM, 1165
rank plus nullity
theorem RPNDD, 1166
scalar multiple
example SMLT, 1167
scalar multiplication
definition LTSM, 1168
spanning range
theorem SSRLT, 1169
sum
example STLT, 1170
surjection
definition SLT, 1171
vector space of, 1172
zero vector
theorem LTTZZ, 1173
linear transformation inverse
via matrix representation
example ILTVR, 1174
linear transformations
compositions
example CTLT, 1175
from matrices
theorem MBLT, 1176
linearly dependent
example LDRN, 1177
via homogeneous system
example LDHS, 1178
linearly dependent columns
Archetype A
example LDCAA, 1179
linearly dependent set
example LDS, 1180
linear combinations within
theorem DLDS, 1181
polynomials
example LDP4, 1182
linearly independent
crazy vector space
example LIC, 1183
extending sets
theorem ELIS, 1184
polynomials
example LIP4, 1185
via homogeneous system
example LIHS, 1186
linearly independent columns
Archetype B
example LICAB, 1187
linearly independent set
example LIS, 1188
example LLDS, 1189
LINM (subsection, section LI), 1190
LINSB (example), 1191
LIP4 (example), 1192
LIS (example), 1193
LISS (section), 1194
LISV (subsection, section LI), 1195
LIVHS (theorem), 1196
LIVRN (theorem), 1197
LLDS (example), 1198
LNS (definition), 1199
LNS (example), 1200
LNS (notation), 1201
LNS (subsection, section FS), 1202
LNSMS (theorem), 1203
LS.MMA (computation, section MMA), 1204
LSMR (definition), 1205
LSMR (notation), 1206
LT (chapter), 1207
LT (definition), 1208
LT (notation), 1209
LT (section), 1210
LT (subsection, section LT), 1211
LTA (definition), 1212
LTC (definition), 1213
LTDB (theorem), 1214
LTDB1 (example), 1215
LTDB2 (example), 1216
LTDB3 (example), 1217
LTLC (subsection, section LT), 1218
LTLC (theorem), 1219
LTM (example), 1220
LTPM (example), 1221
LTPP (example), 1222
LTSM (definition), 1223
LTTZZ (theorem), 1224
M (archetype), 1225
M (chapter), 1226
M (definition), 1227
M (notation), 1228
MA (definition), 1229
MA (example), 1230
MA (notation), 1231
MACN (Property), 1232
mathematica
gram-schmidt (computation), 1233
linear solve (computation), 1234
matrix entry (computation), 1235
matrix inverse (computation), 1236
matrix multiplication (computation), 1237
null space (computation), 1238
row reduce (computation), 1239
transpose of a matrix (computation), 1240
vector form of solutions (computation), 1241
vector linear combinations (computation), 1242
mathematical language
technique L, 1243
matrix
addition
definition MA, 1244
notation, 1245
augmented
definition AM, 1246
column space
definition CSM, 1247
complex conjugate
example CCM, 1248
definition M, 1249
equality
definition ME, 1250
notation, 1251
example AM, 1252
identity
definition IM, 1253
inverse
definition MI, 1254
nonsingular
definition NM, 1255
notation, 1256
of a linear transformation
theorem MLTCV, 1257
product
example PTM, 1258
example PTMEE, 1259
product with vector
definition MVP, 1260
rectangular, 1261
row space
definition RSM, 1262
scalar multiplication
definition MSM, 1263
notation, 1264
singular, 1265
square
definition SQM, 1266
submatrices
example SS, 1267
submatrix
definition SM, 1268
symmetric
definition SYM, 1269
transpose
definition TM, 1270
unitary
definition UM, 1271
unitary is invertible
theorem UMI, 1272
zero
definition ZM, 1273
matrix addition
example MA, 1274
matrix components
notation, 1275
matrix entry
mathematica, 1276
ti83, 1277
ti86, 1278
matrix inverse
Archetype B, 1279
computation
theorem CINM, 1280
mathematica, 1281
nonsingular matrix
theorem NI, 1282
of a matrix inverse
theorem MIMI, 1283
one-sided
theorem OSIS, 1284
product
theorem SS, 1285
scalar multiple
theorem MISM, 1286
size 2 matrices
theorem TTMI, 1287
transpose
theorem MIT, 1288
uniqueness
theorem MIU, 1289
matrix multiplication
associativity
theorem MMA, 1290
complex conjugation
theorem MMCC, 1291
definition MM, 1292
distributivity
theorem MMDAA, 1293
entry-by-entry
theorem EMP, 1294
identity matrix
theorem MMIM, 1295
inner product
theorem MMIP, 1296
mathematica, 1297
noncommutative
example MMNC, 1298
scalar matrix multiplication
theorem MMSMM, 1299
systems of linear equations
theorem SLEMM, 1300
transposes
theorem MMT, 1301
zero matrix
theorem MMZM, 1302
matrix product
as composition of linear transformations
example MPMR, 1303
matrix representation
basis of eigenvectors
example MRBE, 1304
composition of linear transformations
theorem MRCLT, 1305
definition MR, 1306
invertible
theorem IMR, 1307
multiple of a linear transformation
theorem MRMLT, 1308
sum of linear transformations
theorem MRSLT, 1309
theorem FTMR, 1310
matrix representations
converting with change-of-basis
example MRCM, 1311
example OLTTR, 1312
matrix scalar multiplication
example MSM, 1313
matrix vector space
dimension
theorem DM, 1314
matrix-vector product
example MTV, 1315
notation, 1316
MBC (example), 1317
MBLT (theorem), 1318
MC (notation), 1319
MCC (subsection, section MO), 1320
MCCN (Property), 1321
MCN (definition), 1322
MCN (subsection, section CNO), 1323
MCSM (example), 1324
MCT (theorem), 1325
ME (definition), 1326
ME (notation), 1327
ME (subsection, section PEE), 1328
ME (technique, section PT), 1329
ME (theorem), 1330
ME.MMA (computation, section MMA), 1331
ME.TI83 (computation, section TI83), 1332
ME.TI86 (computation, section TI86), 1333
MEASM (subsection, section MO), 1334
MFLT (example), 1335
MI (definition), 1336
MI (example), 1337
MI (notation), 1338
MI.MMA (computation, section MMA), 1339
MICN (Property), 1340
MIMI (theorem), 1341
MINM (section), 1342
MISLE (section), 1343
MISM (theorem), 1344
MIT (theorem), 1345
MIU (theorem), 1346
MIVS (example), 1347
MLT (subsection, section LT), 1348
MLTCV (theorem), 1349
MLTLT (theorem), 1350
MM (definition), 1351
MM (section), 1352
MM (subsection, section MM), 1353
MM.MMA (computation, section MMA), 1354
MMA (section), 1355
MMA (theorem), 1356
MMCC (theorem), 1357
MMDAA (theorem), 1358
MMEE (subsection, section MM), 1359
MMIM (theorem), 1360
MMIP (theorem), 1361
MMNC (example), 1362
MMSMM (theorem), 1363
MMT (theorem), 1364
MMZM (theorem), 1365
MNEM (theorem), 1366
MNSLE (example), 1367
MO (section), 1368
MOLT (example), 1369
more variables than equations
example OSGMD, 1370
theorem CMVEI, 1371
MPMR (example), 1372
MR (definition), 1373
MR (section), 1374
MRBE (example), 1375
MRCB (theorem), 1376
MRCLT (theorem), 1377
MRCM (example), 1378
MRMLT (theorem), 1379
MRS (subsection, section CB), 1380
MRSLT (theorem), 1381
MSCN (example), 1382
MSM (definition), 1383
MSM (example), 1384
MSM (notation), 1385
MTV (example), 1386
multiplicative associativity
complex numbers
Property MACN, 1387
multiplicative commuativity
complex numbers
Property MCCN, 1388
multiplicative inverse
complex numbers
Property MICN, 1389
MVNSE (subsection, section RREF), 1390
MVP (definition), 1391
MVP (notation), 1392
MVP (subsection, section MM), 1393
MVSLD (theorem), 1394
MWIAA (example), 1395
N (archetype), 1396
N (subsection, section O), 1397
N (technique, section PT), 1398
NDMS4 (example), 1399
negation of statements
technique N, 1400
NEM (theorem), 1401
NI (theorem), 1402
NIAO (example), 1403
NIAQ (example), 1404
NIAQR (example), 1405
NIDAU (example), 1406
NKAO (example), 1407
NLT (example), 1408
NLTFO (subsection, section LT), 1409
NM (definition), 1410
NM (example), 1411
NM (section), 1412
NM (subsection, section NM), 1413
NME1 (theorem), 1414
NME2 (theorem), 1415
NME3 (theorem), 1416
NME4 (theorem), 1417
NME5 (theorem), 1418
NME6 (theorem), 1419
NME7 (theorem), 1420
NME8 (theorem), 1421
NME9 (theorem), 1422
NMI (subsection, section MINM), 1423
NMLIC (theorem), 1424
NMPEM (theorem), 1425
NMRRI (theorem), 1426
NMTNS (theorem), 1427
NMUS (theorem), 1428
NOILT (theorem), 1429
NOLT (definition), 1430
NOLT (notation), 1431
NOM (definition), 1432
NOM (notation), 1433
nonsingular
columns as basis
theorem CNMB, 1434
nonsingular matrices
linearly independent columns
theorem NMLIC, 1435
nonsingular matrix
Archetype B
example NM, 1436
column space, 1437
elemntary matrices
theorem NMPEM, 1438
equivalences
theorem NME1, 1439
theorem NME2, 1440
theorem NME3, 1441
theorem NME4, 1442
theorem NME5, 1443
theorem NME6, 1444
theorem NME7, 1445
theorem NME8, 1446
theorem NME9, 1447
matrix inverse, 1448
null space
example NSNM, 1449
nullity, 1450
product of nonsingular matrices
theorem NPNT, 1451
rank
theorem RNNM, 1452
row-reduced
theorem NMRRI, 1453
trivial null space
theorem NMTNS, 1454
unique solutions
theorem NMUS, 1455
nonsingular matrix, row-reduced
example NSR, 1456
norm
example CNSV, 1457
inner product, 1458
notation, 1459
notation
AM, 1460
C, 1461
CCCV, 1462
CCM, 1463
CCN, 1464
CNA, 1465
CNE, 1466
CNM, 1467
CSM, 1468
CV, 1469
CVA, 1470
CVC, 1471
CVE, 1472
CVSM, 1473
D, 1474
DM, 1475
DS, 1476
ELEM, 1477
ES, 1478
IM, 1479
IP, 1480
KLT, 1481
LNS, 1482
LSMR, 1483
LT, 1484
M, 1485
MA, 1486
MC, 1487
ME, 1488
MI, 1489
MSM, 1490
MVP, 1491
NOLT, 1492
NOM, 1493
NSM, 1494
NV, 1495
RLT, 1496
RO, 1497
ROLT, 1498
ROM, 1499
RREFA, 1500
RSM, 1501
SC, 1502
SE, 1503
SETM, 1504
SI, 1505
SM, 1506
SSET, 1507
SSV, 1508
SU, 1509
TM, 1510
VSCV, 1511
VSM, 1512
ZCV, 1513
ZM, 1514
notation for a linear system
example NSE, 1515
NPNT (theorem), 1516
NRFO (subsection, section MR), 1517
NRREF (example), 1518
NS.MMA (computation, section MMA), 1519
NSAO (example), 1520
NSAQ (example), 1521
NSAQR (example), 1522
NSC2A (example), 1523
NSC2S (example), 1524
NSC2Z (example), 1525
NSDAT (example), 1526
NSDS (example), 1527
NSE (example), 1528
NSEAI (example), 1529
NSLE (example), 1530
NSLIL (example), 1531
NSM (definition), 1532
NSM (notation), 1533
NSM (subsection, section HSE), 1534
NSMS (theorem), 1535
NSNM (example), 1536
NSNM (subsection, section NM), 1537
NSR (example), 1538
NSS (example), 1539
NSSLI (subsection, section LI), 1540
Null space
as a span
example NSDS, 1541
null space
Archetype I
example NSEAI, 1542
basis
theorem BNS, 1543
computation
example CNS1, 1544
example CNS2, 1545
isomorphic to kernel, 1546
linearly independent basis
example LINSB, 1547
mathematica, 1548
matrix
definition NSM, 1549
nonsingular matrix, 1550
notation, 1551
singular matrix, 1552
spanning set
example SSNS, 1553
theorem SSNS, 1554
subspace
theorem NSMS, 1555
null space span, linearly independent
Archetype L
example NSLIL, 1556
nullity
computing, 1557
injective linear transformation
theorem NOILT, 1558
linear transformation
definition NOLT, 1559
matrix, 1560
definition NOM, 1561
notation, 1562, 1563
square matrix, 1564
NV (definition), 1565
NV (notation), 1566
O (archetype), 1567
O (Property), 1568
O (section), 1569
OBC (subsection, section B), 1570
OC (Property), 1571
OCN (Property), 1572
OD (subsection, section SD), 1573
OLTTR (example), 1574
OM (Property), 1575
one
column vectors
Property OC, 1576
complex numbers
Property OCN, 1577
matrices
Property OM, 1578
vectors
Property O, 1579
ONFV (example), 1580
ONS (definition), 1581
ONTV (example), 1582
orthogonal
linear independence
theorem OSLI, 1583
set
example AOS, 1584
set of vectors
definition OSV, 1585
vector pairs
definition OV, 1586
orthogonal vectors
example TOV, 1587
orthonormal
definition ONS, 1588
matrix columns
example OSMC, 1589
orthonormal set
four vectors
example ONFV, 1590
three vectors
example ONTV, 1591
OSGMD (example), 1592
OSIS (theorem), 1593
OSLI (theorem), 1594
OSMC (example), 1595
OSV (definition), 1596
OV (definition), 1597
OV (subsection, section O), 1598
P (appendix), 1599
P (archetype), 1600
P (technique, section PT), 1601
particular solutions
example PSHS, 1602
PCNA (theorem), 1603
PCVS (example), 1604
PD (section), 1605
PDM (section), 1606
PEE (section), 1607
PEEF (theorem), 1608
PI (definition), 1609
PI (subsection, section LT), 1610
PI (technique, section PT), 1611
PIP (theorem), 1612
PM (example), 1613
PM (subsection, section EE), 1614
PMI (subsection, section MISLE), 1615
PMM (subsection, section MM), 1616
PMR (subsection, section MR), 1617
polynomial
of a matrix
example PM, 1618
polynomial vector space
dimension
theorem DP, 1619
practice
technique P, 1620
pre-image
definition PI, 1621
kernel
theorem KPI, 1622
pre-images
example SPIAS, 1623
Property
AA, 1624
AAC, 1625
AACN, 1626
AAM, 1627
AC, 1628
ACC, 1629
ACCN, 1630
ACM, 1631
AI, 1632
AIC, 1633
AICN, 1634
AIM, 1635
C, 1636
CC, 1637
CM, 1638
DCN, 1639
DMAM, 1640
DSA, 1641
DSAC, 1642
DSAM, 1643
DVA, 1644
DVAC, 1645
MACN, 1646
MCCN, 1647
MICN, 1648
O, 1649
OC, 1650
OCN, 1651
OM, 1652
SC, 1653
SCC, 1654
SCM, 1655
SMA, 1656
SMAC, 1657
SMAM, 1658
Z, 1659
ZC, 1660
ZCN, 1661
ZM, 1662
PSHS (example), 1663
PSHS (subsection, section LC), 1664
PSM (subsection, section SD), 1665
PSPHS (theorem), 1666
PSS (subsection, section SSLE), 1667
PSSD (theorem), 1668
PSSLS (theorem), 1669
PT (section), 1670
PTM (example), 1671
PTMEE (example), 1672
Q (archetype), 1673
R (archetype), 1674
R (chapter), 1675
range
full
example FRAN, 1676
isomorphic to column space
theorem RCSI, 1677
linear transformation
example RAO, 1678
notation, 1679
of a linear transformation
definition RLT, 1680
pre-image
theorem RPI, 1681
subspace
theorem RLTS, 1682
surjective linear transformation
theorem RSLT, 1683
via matrix representation
example RVMR, 1684
rank
computing
theorem CRN, 1685
linear transformation
definition ROLT, 1686
matrix
definition ROM, 1687
example RNM, 1688
notation, 1689, 1690
of transpose
example RRTI, 1691
square matrix
example RNSM, 1692
surjective linear transformation
theorem ROSLT, 1693
transpose
theorem RMRT, 1694
rank+nullity
theorem RPNC, 1695
RAO (example), 1696
RCLS (theorem), 1697
RCSI (theorem), 1698
RD (subsection, section VS), 1699
READ (subsection, section B), 1700
READ (subsection, section CB), 1701
READ (subsection, section CRS), 1702
READ (subsection, section D), 1703
READ (subsection, section DM), 1704
READ (subsection, section EE), 1705
READ (subsection, section FS), 1706
READ (subsection, section HSE), 1707
READ (subsection, section ILT), 1708
READ (subsection, section IVLT), 1709
READ (subsection, section LC), 1710
READ (subsection, section LDS), 1711
READ (subsection, section LI), 1712
READ (subsection, section LISS), 1713
READ (subsection, section LT), 1714
READ (subsection, section MINM), 1715
READ (subsection, section MISLE), 1716
READ (subsection, section MM), 1717
READ (subsection, section MO), 1718
READ (subsection, section MR), 1719
READ (subsection, section NM), 1720
READ (subsection, section O), 1721
READ (subsection, section PD), 1722
READ (subsection, section PDM), 1723
READ (subsection, section PEE), 1724
READ (subsection, section RREF), 1725
READ (subsection, section S), 1726
READ (subsection, section SD), 1727
READ (subsection, section SLT), 1728
READ (subsection, section SS), 1729
READ (subsection, section SSLE), 1730
READ (subsection, section TSS), 1731
READ (subsection, section VO), 1732
READ (subsection, section VR), 1733
READ (subsection, section VS), 1734
READ (subsection, section WILA), 1735
reduced row-echelon form
analysis
notation, 1736
definition RREF, 1737
example NRREF, 1738
example RREF, 1739
extended
definition EEF, 1740
notation
example RREFN, 1741
unique
theorem RREFU, 1742
reducing a span
example RSC5, 1743
relation of linear dependence
definition RLD, 1744
definition RLDCV, 1745
REM (definition), 1746
REMEF (theorem), 1747
REMES (theorem), 1748
REMRS (theorem), 1749
RES (example), 1750
RLD (definition), 1751
RLDCV (definition), 1752
RLT (definition), 1753
RLT (notation), 1754
RLT (subsection, section SLT), 1755
RLTS (theorem), 1756
RMRT (theorem), 1757
RNLT (subsection, section IVLT), 1758
RNM (example), 1759
RNM (subsection, section D), 1760
RNNM (subsection, section D), 1761
RNNM (theorem), 1762
RNSM (example), 1763
RO (definition), 1764
RO (notation), 1765
RO (subsection, section RREF), 1766
ROLT (definition), 1767
ROLT (notation), 1768
ROM (definition), 1769
ROM (notation), 1770
ROSLT (theorem), 1771
row operations
definition RO, 1772
elementary matrices, 1773, 1774
notation, 1775
row reduce
mathematica, 1776
ti83, 1777
ti86, 1778
row space
Archetype I
example RSAI, 1779
as column space, 1780
basis
example RSB, 1781
theorem BRS, 1782
matrix, 1783
notation, 1784
row-equivalent matrices
theorem REMRS, 1785
subspace
theorem RSMS, 1786
row-equivalent matrices
definition REM, 1787
example TREM, 1788
row space, 1789
row spaces
example RSREM, 1790
theorem REMES, 1791
row-reduce
the verb
definition RR, 1792
row-reduced matrices
theorem REMEF, 1793
RPI (theorem), 1794
RPNC (theorem), 1795
RPNDD (theorem), 1796
RR (definition), 1797
RR.MMA (computation, section MMA), 1798
RR.TI83 (computation, section TI83), 1799
RR.TI86 (computation, section TI86), 1800
RREF (definition), 1801
RREF (example), 1802
RREF (section), 1803
RREF (subsection, section RREF), 1804
RREFA (notation), 1805
RREFN (example), 1806
RREFU (theorem), 1807
RRTI (example), 1808
RS (example), 1809
RSAI (example), 1810
RSB (example), 1811
RSC5 (example), 1812
RSLT (theorem), 1813
RSM (definition), 1814
RSM (notation), 1815
RSM (subsection, section CRS), 1816
RSMS (theorem), 1817
RSNS (example), 1818
RSREM (example), 1819
RSSC4 (example), 1820
RT (subsection, section PD), 1821
RVMR (example), 1822
S (archetype), 1823
S (definition), 1824
S (example), 1825
S (section), 1826
SAA (example), 1827
SAB (example), 1828
SABMI (example), 1829
SAE (example), 1830
SAN (example), 1831
SAR (example), 1832
SAV (example), 1833
SC (definition), 1834
SC (example), 1835
SC (notation), 1836
SC (Property), 1837
SC (subsection, section S), 1838
SC (subsection, section SET), 1839
SC3 (example), 1840
SCAA (example), 1841
SCAB (example), 1842
SCAD (example), 1843
scalar closure
column vectors
Property SCC, 1844
matrices
Property SCM, 1845
vectors
Property SC, 1846
scalar multiple
matrix inverse, 1847
scalar multiplication
canceling scalars
theorem CSSM, 1848
canceling vectors
theorem CVSM, 1849
zero scalar
theorem ZSSM, 1850
zero vector
theorem ZVSM, 1851
zero vector result
theorem SMEZV, 1852
scalar multiplication associativity
column vectors
Property SMAC, 1853
matrices
Property SMAM, 1854
vectors
Property SMA, 1855
SCB (theorem), 1856
SCC (Property), 1857
SCM (Property), 1858
SD (section), 1859
SE (definition), 1860
SE (notation), 1861
SEE (example), 1862
SEEF (example), 1863
SER (theorem), 1864
set
cardinality
definition C, 1865
example CS, 1866
notation, 1867
complement
definition SC, 1868
example SC, 1869
notation, 1870
definition SET, 1871
empty
definition ES, 1872
equality
definition SE, 1873
notation, 1874
intersection
definition SI, 1875
example SI, 1876
notation, 1877
membership
example SETM, 1878
notation, 1879
size, 1880
subset, 1881
union
definition SU, 1882
example SU, 1883
notation, 1884
SET (definition), 1885
SET (section), 1886
SETM (example), 1887
SETM (notation), 1888
shoes, 1889
SHS (subsection, section HSE), 1890
SI (definition), 1891
SI (example), 1892
SI (notation), 1893
SI (subsection, section IVLT), 1894
SIM (definition), 1895
similar matrices
equal eigenvalues
example EENS, 1896
eual eigenvalues
theorem SMEE, 1897
example SMS3, 1898
example SMS5, 1899
similarity
definition SIM, 1900
equivalence relation
theorem SER, 1901
singular matrix
Archetype A
example S, 1902
null space
example NSS, 1903
singular matrix, row-reduced
example SRR, 1904
SLE (chapter), 1905
SLE (definition), 1906
SLE (subsection, section SSLE), 1907
SLELT (subsection, section IVLT), 1908
SLEMM (theorem), 1909
SLSLC (theorem), 1910
SLT (definition), 1911
SLT (section), 1912
SLTB (theorem), 1913
SLTD (subsection, section SLT), 1914
SLTD (theorem), 1915
SLTLT (theorem), 1916
SM (definition), 1917
SM (notation), 1918
SM (subsection, section SD), 1919
SM32 (example), 1920
SMA (Property), 1921
SMAC (Property), 1922
SMAM (Property), 1923
SMEE (theorem), 1924
SMEZV (theorem), 1925
SMLT (example), 1926
SMS (theorem), 1927
SMS3 (example), 1928
SMS5 (example), 1929
SMZD (theorem), 1930
SMZE (theorem), 1931
SNCM (theorem), 1932
SO (subsection, section SET), 1933
socks, 1934
SOL (subsection, section B), 1935
SOL (subsection, section CB), 1936
SOL (subsection, section CRS), 1937
SOL (subsection, section D), 1938
SOL (subsection, section DM), 1939
SOL (subsection, section EE), 1940
SOL (subsection, section FS), 1941
SOL (subsection, section HSE), 1942
SOL (subsection, section ILT), 1943
SOL (subsection, section IVLT), 1944
SOL (subsection, section LC), 1945
SOL (subsection, section LDS), 1946
SOL (subsection, section LI), 1947
SOL (subsection, section LISS), 1948
SOL (subsection, section LT), 1949
SOL (subsection, section MINM), 1950
SOL (subsection, section MISLE), 1951
SOL (subsection, section MM), 1952
SOL (subsection, section MO), 1953
SOL (subsection, section MR), 1954
SOL (subsection, section NM), 1955
SOL (subsection, section PD), 1956
SOL (subsection, section PDM), 1957
SOL (subsection, section PEE), 1958
SOL (subsection, section RREF), 1959
SOL (subsection, section S), 1960
SOL (subsection, section SD), 1961
SOL (subsection, section SLT), 1962
SOL (subsection, section SS), 1963
SOL (subsection, section SSLE), 1964
SOL (subsection, section TSS), 1965
SOL (subsection, section VO), 1966
SOL (subsection, section VR), 1967
SOL (subsection, section WILA), 1968
solution set
Archetype A
example SAA, 1969
archetype E
example SAE, 1970
theorem PSPHS, 1971
solution sets
possibilities
theorem PSSLS, 1972
solution vector
definition SV, 1973
solving homogeneous system
Archetype A
example HISAA, 1974
Archetype B
example HUSAB, 1975
Archetype D
example HISAD, 1976
solving nonlinear equations
example STNE, 1977
SP4 (example), 1978
span
basic
example ABS, 1979
basis
theorem BS, 1980
definition SS, 1981
definition SSCV, 1982
improved
example IAS, 1983
notation, 1984
reducing
example RSSC4, 1985
reduction
example RS, 1986
removing vectors
example COV, 1987
reworking elements
example RES, 1988
set of polynomials
example SSP, 1989
subspace
theorem SSS, 1990
span of columns
Archetype A
example SCAA, 1991
Archetype B
example SCAB, 1992
Archetype D
example SCAD, 1993
spanning set
crazy vector space
example SSC, 1994
definition TSVS, 1995
matrices
example SSM22, 1996
more vectors
theorem SSLD, 1997
polynomials
example SSP4, 1998
SPIAS (example), 1999
SQM (definition), 2000
SRR (example), 2001
SS (definition), 2002
SS (example), 2003
SS (section), 2004
SS (subsection, section LISS), 2005
SS (theorem), 2006
SSC (example), 2007
SSCV (definition), 2008
SSET (definition), 2009
SSET (example), 2010
SSET (notation), 2011
SSLD (theorem), 2012
SSLE (section), 2013
SSM22 (example), 2014
SSNS (example), 2015
SSNS (subsection, section SS), 2016
SSNS (theorem), 2017
SSP (example), 2018
SSP4 (example), 2019
SSRLT (theorem), 2020
SSS (theorem), 2021
SSSLT (subsection, section SLT), 2022
SSV (notation), 2023
SSV (subsection, section SS), 2024
starting proofs
technique GS, 2025
STLT (example), 2026
STNE (example), 2027
SU (definition), 2028
SU (example), 2029
SU (notation), 2030
submatrix
notation, 2031
subset
definition SSET, 2032
notation, 2033
subspace
as null space
example RSNS, 2034
characterized
example ASC, 2035
definition S, 2036
in
example SP4, 2037
not, additive closure
example NSC2A, 2038
not, scalar closure
example NSC2S, 2039
not, zero vector
example NSC2Z, 2040
testing
theorem TSS, 2041
trivial
definition TS, 2042
verification
example SC3, 2043
example SM32, 2044
subspaces
equal dimension
theorem EDYES, 2045
surjective
Archetype N
example SAN, 2046
example SAR, 2047
not
example NSAQ, 2048
example NSAQR, 2049
not, Archetype O
example NSAO, 2050
not, by dimension
example NSDAT, 2051
polynomials to matrices
example SAV, 2052
surjective linear transformation
bases
theorem SLTB, 2053
surjective linear transformations
dimension
theorem SLTD, 2054
SUV (definition), 2055
SUVB (theorem), 2056
SUVOS (example), 2057
SV (definition), 2058
SVP4 (example), 2059
SYM (definition), 2060
SYM (example), 2061
symmetric matrices
theorem SMS, 2062
symmetric matrix
example SYM, 2063
system of equations
vector equality
example VESE, 2064
system of linear equations
definition SLE, 2065
T (archetype), 2066
T (part), 2067
T (technique, section PT), 2068
TCSD (example), 2069
technique
C, 2070
CD, 2071
CP, 2072
CV, 2073
D, 2074
DC, 2075
E, 2076
GS, 2077
I, 2078
L, 2079
LC, 2080
ME, 2081
N, 2082
P, 2083
PI, 2084
T, 2085
U, 2086
theorem
AISM, 2087
AIU, 2088
BCS, 2089
BIS, 2090
BNS, 2091
BRS, 2092
BS, 2093
CB, 2094
CCRA, 2095
CCRM, 2096
CCT, 2097
CFDVS, 2098
CILTI, 2099
CINM, 2100
CIVLT, 2101
CLI, 2102
CLTLT, 2103
CMVEI, 2104
CNMB, 2105
COB, 2106
CRMA, 2107
CRMSM, 2108
CRN, 2109
CRSM, 2110
CRVA, 2111
CSCS, 2112
CSLTS, 2113
CSMS, 2114
CSNM, 2115
CSRN, 2116
CSRST, 2117
CSS, 2118
CSSM, 2119
CUMOS, 2120
CVSM, 2121
DC, 2122
DCM, 2123
DCP, 2124
DEC, 2125
DED, 2126
DEM, 2127
DEMMM, 2128
DER, 2129
DERC, 2130
DFS, 2131
DIM, 2132
DLDS, 2133
DM, 2134
DMFE, 2135
DMST, 2136
DP, 2137
DRCM, 2138
DRCMA, 2139
DRCS, 2140
DRMM, 2141
DT, 2142
DZRC, 2143
EB, 2144
EDELI, 2145
EDYES, 2146
EER, 2147
EIM, 2148
ELIS, 2149
EMDRO, 2150
EMHE, 2151
EMMVP, 2152
EMN, 2153
EMNS, 2154
EMP, 2155
EMRCP, 2156
EMS, 2157
EOMP, 2158
EOPSS, 2159
EPM, 2160
ERMCP, 2161
ESMM, 2162
ETM, 2163
FS, 2164
FTMR, 2165
FVCS, 2166
G, 2167
GSPCV, 2168
HMOE, 2169
HMRE, 2170
HMVEI, 2171
HSC, 2172
ICBM, 2173
ICLT, 2174
IFDVS, 2175
IILT, 2176
ILTB, 2177
ILTD, 2178
ILTIS, 2179
ILTLI, 2180
ILTLT, 2181
IMILT, 2182
IMR, 2183
IPAC, 2184
IPN, 2185
IPSM, 2186
IPVA, 2187
ISRN, 2188
IVSED, 2189
KILT, 2190
KLTS, 2191
KNSI, 2192
KPI, 2193
LIVHS, 2194
LIVRN, 2195
LNSMS, 2196
LTDB, 2197
LTLC, 2198
LTTZZ, 2199
MBLT, 2200
MCT, 2201
ME, 2202
MIMI, 2203
MISM, 2204
MIT, 2205
MIU, 2206
MLTCV, 2207
MLTLT, 2208
MMA, 2209
MMCC, 2210
MMDAA, 2211
MMIM, 2212
MMIP, 2213
MMSMM, 2214
MMT, 2215
MMZM, 2216
MNEM, 2217
MRCB, 2218
MRCLT, 2219
MRMLT, 2220
MRSLT, 2221
MVSLD, 2222
NEM, 2223
NI, 2224
NME1, 2225
NME2, 2226
NME3, 2227
NME4, 2228
NME5, 2229
NME6, 2230
NME7, 2231
NME8, 2232
NME9, 2233
NMLIC, 2234
NMPEM, 2235
NMRRI, 2236
NMTNS, 2237
NMUS, 2238
NOILT, 2239
NPNT, 2240
NSMS, 2241
OSIS, 2242
OSLI, 2243
PCNA, 2244
PEEF, 2245
PIP, 2246
PSPHS, 2247
PSSD, 2248
PSSLS, 2249
RCLS, 2250
RCSI, 2251
REMEF, 2252
REMES, 2253
REMRS, 2254
RLTS, 2255
RMRT, 2256
RNNM, 2257
ROSLT, 2258
RPI, 2259
RPNC, 2260
RPNDD, 2261
RREFU, 2262
RSLT, 2263
RSMS, 2264
SCB, 2265
SER, 2266
SLEMM, 2267
SLSLC, 2268
SLTB, 2269
SLTD, 2270
SLTLT, 2271
SMEE, 2272
SMEZV, 2273
SMS, 2274
SMZD, 2275
SMZE, 2276
SNCM, 2277
SS, 2278
SSLD, 2279
SSNS, 2280
SSRLT, 2281
SSS, 2282
SUVB, 2283
technique T, 2284
TMA, 2285
TMSM, 2286
TSS, 2287
TT, 2288
TTMI, 2289
UMI, 2290
UMPIP, 2291
VAC, 2292
VFSLS, 2293
VRI, 2294
VRILT, 2295
VRLT, 2296
VRRB, 2297
VRS, 2298
VSLT, 2299
VSPCV, 2300
VSPM, 2301
ZSSM, 2302
ZVSM, 2303
ZVU, 2304
ti83
matrix entry (computation), 2305
row reduce (computation), 2306
vector linear combinations (computation), 2307
TI83 (section), 2308
ti86
matrix entry (computation), 2309
row reduce (computation), 2310
transpose of a matrix (computation), 2311
vector linear combinations (computation), 2312
TI86 (section), 2313
TIVS (example), 2314
TKAP (example), 2315
TLC (example), 2316
TM (definition), 2317
TM (example), 2318
TM (notation), 2319
TM.MMA (computation, section MMA), 2320
TM.TI86 (computation, section TI86), 2321
TMA (theorem), 2322
TMP (example), 2323
TMSM (theorem), 2324
TOV (example), 2325
trail mix
example TMP, 2326
transpose
matrix scalar multiplication
theorem TMSM, 2327
example TM, 2328
matrix addition
theorem TMA, 2329
matrix inverse, 2330, 2331
notation, 2332
scalar multiplication, 2333
transpose of a matrix
mathematica, 2334
ti86, 2335
transpose of a transpose
theorem TT, 2336
TREM (example), 2337
trivial solution
system of equations
definition TSHSE, 2338
TS (definition), 2339
TS (subsection, section S), 2340
TSHSE (definition), 2341
TSM (subsection, section MO), 2342
TSS (section), 2343
TSS (subsection, section S), 2344
TSS (theorem), 2345
TSVS (definition), 2346
TT (theorem), 2347
TTMI (theorem), 2348
TTS (example), 2349
typical systems,
example TTS, 2350
U (archetype), 2351
U (technique, section PT), 2352
UM (definition), 2353
UM (subsection, section MINM), 2354
UM3 (example), 2355
UMI (theorem), 2356
UMPIP (theorem), 2357
unique solution,
example US, 2358
example USR, 2359
uniqueness
technique U, 2360
unit vectors
basis
theorem SUVB, 2361
definition SUV, 2362
orthogonal
example SUVOS, 2363
unitary
permutation matrix
example UPM, 2364
size 3
example UM3, 2365
unitary matrices
columns
theorem CUMOS, 2366
unitary matrix
inner product
theorem UMPIP, 2367
UPM (example), 2368
URREF (subsection, section LC), 2369
US (example), 2370
USR (example), 2371
V (archetype), 2372
V (chapter), 2373
VA (example), 2374
VAC (theorem), 2375
VEASM (subsection, section VO), 2376
vector
addition
definition CVA, 2377
column
definition CV, 2378
equality
definition CVE, 2379
notation, 2380
inner product
definition IP, 2381
norm
definition NV, 2382
notation, 2383
of constants
definition VOC, 2384
product with matrix, 2385, 2386
scalar multiplication
definition CVSM, 2387
vector addition
example VA, 2388
vector component
notation, 2389
vector form of solutions
Archetype D
example VFSAD, 2390
Archetype I
example VFSAI, 2391
Archetype L
example VFSAL, 2392
example VFS, 2393
mathematica, 2394
theorem VFSLS, 2395
vector linear combinations
mathematica, 2396
ti83, 2397
ti86, 2398
vector representation
example AVR, 2399
example VRC4, 2400
injective
theorem VRI, 2401
invertible
theorem VRILT, 2402
linear transformation
definition VR, 2403
theorem VRLT, 2404
surjective
theorem VRS, 2405
theorem VRRB, 2406
vector representations
polynomials
example VRP2, 2407
vector scalar multiplication
example CVSM, 2408
vector space
characterization
theorem CFDVS, 2409
column vectors
definition VSCV, 2410
definition VS, 2411
infinite dimension
example VSPUD, 2412
linear transformations
theorem VSLT, 2413
vector space of column vectors
notation, 2414
vector space of functions
example VSF, 2415
vector space of infinite sequences
example VSIS, 2416
vector space of matrices
definition VSM, 2417
example VSM, 2418
notation, 2419
vector space of polynomials
example VSP, 2420
vector space properties
column vectors
theorem VSPCV, 2421
matrices
theorem VSPM, 2422
vector space, crazy
example CVS, 2423
vector space, singleton
example VSS, 2424
vector spaces
isomorphic
definition IVS, 2425
theorem IFDVS, 2426
VESE (example), 2427
VFS (example), 2428
VFSAD (example), 2429
VFSAI (example), 2430
VFSAL (example), 2431
VFSLS (theorem), 2432
VFSS (subsection, section LC), 2433
VFSS.MMA (computation, section MMA), 2434
VLC.MMA (computation, section MMA), 2435
VLC.TI83 (computation, section TI83), 2436
VLC.TI86 (computation, section TI86), 2437
VO (section), 2438
VOC (definition), 2439
VR (definition), 2440
VR (section), 2441
VR (subsection, section LISS), 2442
VRC4 (example), 2443
VRI (theorem), 2444
VRILT (theorem), 2445
VRLT (theorem), 2446
VRP2 (example), 2447
VRRB (theorem), 2448
VRS (theorem), 2449
VS (chapter), 2450
VS (definition), 2451
VS (section), 2452
VS (subsection, section VS), 2453
VSCV (definition), 2454
VSCV (example), 2455
VSCV (notation), 2456
VSF (example), 2457
VSIS (example), 2458
VSLT (theorem), 2459
VSM (definition), 2460
VSM (example), 2461
VSM (notation), 2462
VSP (example), 2463
VSP (subsection, section MO), 2464
VSP (subsection, section VO), 2465
VSP (subsection, section VS), 2466
VSPCV (theorem), 2467
VSPM (theorem), 2468
VSPUD (example), 2469
VSS (example), 2470
W (archetype), 2471
WILA (section), 2472
X (archetype), 2473
Z (Property), 2474
ZC (Property), 2475
ZCN (Property), 2476
ZCV (definition), 2477
ZCV (notation), 2478
zero
complex numbers
Property ZCN, 2479
zero column vector
definition ZCV, 2480
notation, 2481
zero matrix
notation, 2482
zero vector
column vectors
Property ZC, 2483
matrices
Property ZM, 2484
unique
theorem ZVU, 2485
vectors
Property Z, 2486
ZM (definition), 2487
ZM (notation), 2488
ZM (Property), 2489
ZNDAB (example), 2490
ZSSM (theorem), 2491
ZVSM (theorem), 2492
ZVU (theorem), 2493