Robert Beezer
It has numerous worked examples and exercises, along with precise statements of definitions and complete proofs of every theorem, making it ideal for independent study. Distributed under the open-content GNU Free Documentation License (GFDL), evaluation copies, an online version and updates are available at the book’s website, linear.ups.edu.
This edition was published in August 2008 in several formats. Please visit the book’s storefront to check for updated editions and explantions of the formats formats prior to ordering.
Table of contents :
Table of Contents……Page 7
Contributors……Page 17
Definitions……Page 19
Theorems……Page 25
Notation……Page 33
Figures……Page 35
Examples……Page 37
Preface……Page 45
Acknowledgements……Page 51
LA “Linear” + “Algebra”……Page 55
AA An Application……Page 56
READ Reading Questions……Page 59
EXC Exercises……Page 60
SOL Solutions……Page 61
SLE Systems of Linear Equations……Page 63
PSS Possibilities for Solution Sets……Page 64
ESEO Equivalent Systems and Equation Operations……Page 65
READ Reading Questions……Page 70
EXC Exercises……Page 71
SOL Solutions……Page 73
MVNSE Matrix and Vector Notation for Systems of Equations……Page 77
RO Row Operations……Page 80
RREF Reduced Row-Echelon Form……Page 82
READ Reading Questions……Page 92
EXC Exercises……Page 93
SOL Solutions……Page 97
CS Consistent Systems……Page 103
FV Free Variables……Page 108
READ Reading Questions……Page 110
EXC Exercises……Page 111
SOL Solutions……Page 113
SHS Solutions of Homogeneous Systems……Page 115
NSM Null Space of a Matrix……Page 117
READ Reading Questions……Page 119
EXC Exercises……Page 120
SOL Solutions……Page 122
NM Nonsingular Matrices……Page 125
NSNM Null Space of a Nonsingular Matrix……Page 127
READ Reading Questions……Page 129
EXC Exercises……Page 130
SOL Solutions……Page 132
SLE Systems of Linear Equations……Page 137
VEASM Vector Equality, Addition, Scalar Multiplication……Page 139
VSP Vector Space Properties……Page 142
READ Reading Questions……Page 143
EXC Exercises……Page 144
SOL Solutions……Page 145
LC Linear Combinations……Page 147
VFSS Vector Form of Solution Sets……Page 151
PSHS Particular Solutions, Homogeneous Solutions……Page 161
READ Reading Questions……Page 163
EXC Exercises……Page 165
SOL Solutions……Page 167
SSV Span of a Set of Vectors……Page 169
SSNS Spanning Sets of Null Spaces……Page 174
READ Reading Questions……Page 178
EXC Exercises……Page 179
SOL Solutions……Page 181
LISV Linearly Independent Sets of Vectors……Page 187
LINM Linear Independence and Nonsingular Matrices……Page 192
NSSLI Null Spaces, Spans, Linear Independence……Page 193
READ Reading Questions……Page 196
EXC Exercises……Page 197
SOL Solutions……Page 201
LDSS Linearly Dependent Sets and Spans……Page 207
COV Casting Out Vectors……Page 209
READ Reading Questions……Page 215
EXC Exercises……Page 216
SOL Solutions……Page 218
CAV Complex Arithmetic and Vectors……Page 221
IP Inner products……Page 222
N Norm……Page 224
OV Orthogonal Vectors……Page 226
GSP Gram-Schmidt Procedure……Page 228
READ Reading Questions……Page 232
EXC Exercises……Page 233
SOL Solutions……Page 234
V Vectors……Page 235
MEASM Matrix Equality, Addition, Scalar Multiplication……Page 237
VSP Vector Space Properties……Page 238
TSM Transposes and Symmetric Matrices……Page 240
MCC Matrices and Complex Conjugation……Page 242
AM Adjoint of a Matrix……Page 243
READ Reading Questions……Page 245
EXC Exercises……Page 246
SOL Solutions……Page 248
MVP Matrix-Vector Product……Page 249
MM Matrix Multiplication……Page 252
MMEE Matrix Multiplication, Entry-by-Entry……Page 253
PMM Properties of Matrix Multiplication……Page 254
HM Hermitian Matrices……Page 259
READ Reading Questions……Page 260
EXC Exercises……Page 261
SOL Solutions……Page 263
MISLE Matrix Inverses and Systems of Linear Equations……Page 265
IM Inverse of a Matrix……Page 266
CIM Computing the Inverse of a Matrix……Page 267
PMI Properties of Matrix Inverses……Page 271
READ Reading Questions……Page 274
EXC Exercises……Page 275
SOL Solutions……Page 277
NMI Nonsingular Matrices are Invertible……Page 279
UM Unitary Matrices……Page 282
READ Reading Questions……Page 285
EXC Exercises……Page 286
SOL Solutions……Page 287
CSSE Column Spaces and Systems of Equations……Page 289
CSSOC Column Space Spanned by Original Columns……Page 292
CSNM Column Space of a Nonsingular Matrix……Page 294
RSM Row Space of a Matrix……Page 295
READ Reading Questions……Page 301
EXC Exercises……Page 302
SOL Solutions……Page 306
LNS Left Null Space……Page 311
CRS Computing Column Spaces……Page 312
EEF Extended echelon form……Page 314
FS Four Subsets……Page 317
READ Reading Questions……Page 325
EXC Exercises……Page 326
SOL Solutions……Page 328
M Matrices……Page 333
VS Vector Spaces……Page 335
EVS Examples of Vector Spaces……Page 336
VSP Vector Space Properties……Page 341
READ Reading Questions……Page 344
EXC Exercises……Page 346
SOL Solutions……Page 347
S Subspaces……Page 349
TS Testing Subspaces……Page 350
TSS The Span of a Set……Page 354
SC Subspace Constructions……Page 359
READ Reading Questions……Page 360
EXC Exercises……Page 361
SOL Solutions……Page 362
LI Linear Independence……Page 365
SS Spanning Sets……Page 369
VR Vector Representation……Page 373
READ Reading Questions……Page 374
EXC Exercises……Page 375
SOL Solutions……Page 377
B Bases……Page 381
BSCV Bases for Spans of Column Vectors……Page 384
BNM Bases and Nonsingular Matrices……Page 386
OBC Orthonormal Bases and Coordinates……Page 387
READ Reading Questions……Page 391
EXC Exercises……Page 392
SOL Solutions……Page 393
D Dimension……Page 395
DVS Dimension of Vector Spaces……Page 398
RNM Rank and Nullity of a Matrix……Page 400
RNNM Rank and Nullity of a Nonsingular Matrix……Page 402
READ Reading Questions……Page 403
EXC Exercises……Page 404
SOL Solutions……Page 406
GT Goldilocks’ Theorem……Page 409
RT Ranks and Transposes……Page 412
DFS Dimension of Four Subspaces……Page 413
DS Direct Sums……Page 414
READ Reading Questions……Page 419
EXC Exercises……Page 420
SOL Solutions……Page 421
VS Vector Spaces……Page 423
EM Elementary Matrices……Page 425
DD Definition of the Determinant……Page 429
CD Computing Determinants……Page 431
READ Reading Questions……Page 434
EXC Exercises……Page 435
SOL Solutions……Page 436
DRO Determinants and Row Operations……Page 437
DROEM Determinants, Row Operations, Elementary Matrices……Page 441
DNMMM Determinants, Nonsingular Matrices, Matrix Multiplication……Page 443
READ Reading Questions……Page 445
EXC Exercises……Page 446
SOL Solutions……Page 447
D Determinants……Page 449
EEM Eigenvalues and Eigenvectors of a Matrix……Page 451
PM Polynomials and Matrices……Page 453
EEE Existence of Eigenvalues and Eigenvectors……Page 454
CEE Computing Eigenvalues and Eigenvectors……Page 457
ECEE Examples of Computing Eigenvalues and Eigenvectors……Page 460
READ Reading Questions……Page 467
EXC Exercises……Page 468
SOL Solutions……Page 469
PEE Properties of Eigenvalues and Eigenvectors……Page 473
ME Multiplicities of Eigenvalues……Page 478
EHM Eigenvalues of Hermitian Matrices……Page 481
READ Reading Questions……Page 482
EXC Exercises……Page 483
SOL Solutions……Page 484
SM Similar Matrices……Page 487
PSM Properties of Similar Matrices……Page 488
D Diagonalization……Page 490
FS Fibonacci Sequences……Page 497
READ Reading Questions……Page 499
EXC Exercises……Page 500
SOL Solutions……Page 501
E Eigenvalues……Page 505
LT Linear Transformations……Page 507
LTC Linear Transformation Cartoons……Page 511
MLT Matrices and Linear Transformations……Page 512
LTLC Linear Transformations and Linear Combinations……Page 516
PI Pre-Images……Page 519
NLTFO New Linear Transformations From Old……Page 521
READ Reading Questions……Page 525
EXC Exercises……Page 526
SOL Solutions……Page 528
EILT Examples of Injective Linear Transformations……Page 531
KLT Kernel of a Linear Transformation……Page 534
ILTLI Injective Linear Transformations and Linear Independence……Page 539
CILT Composition of Injective Linear Transformations……Page 540
READ Reading Questions……Page 541
EXC Exercises……Page 542
SOL Solutions……Page 544
ESLT Examples of Surjective Linear Transformations……Page 547
RLT Range of a Linear Transformation……Page 550
SSSLT Spanning Sets and Surjective Linear Transformations……Page 555
CSLT Composition of Surjective Linear Transformations……Page 557
READ Reading Questions……Page 558
EXC Exercises……Page 559
SOL Solutions……Page 561
IVLT Invertible Linear Transformations……Page 563
IV Invertibility……Page 566
SI Structure and Isomorphism……Page 569
RNLT Rank and Nullity of a Linear Transformation……Page 571
SLELT Systems of Linear Equations and Linear Transformations……Page 574
READ Reading Questions……Page 576
EXC Exercises……Page 577
SOL Solutions……Page 579
LT Linear Transformations……Page 583
VR Vector Representations……Page 585
CVS Characterization of Vector Spaces……Page 590
CP Coordinatization Principle……Page 591
READ Reading Questions……Page 593
EXC Exercises……Page 595
SOL Solutions……Page 596
MR Matrix Representations……Page 597
NRFO New Representations from Old……Page 603
PMR Properties of Matrix Representations……Page 607
IVLT Invertible Linear Transformations……Page 612
READ Reading Questions……Page 615
EXC Exercises……Page 616
SOL Solutions……Page 619
EELT Eigenvalues and Eigenvectors of Linear Transformations……Page 627
CBM Change-of-Basis Matrix……Page 628
MRS Matrix Representations and Similarity……Page 633
CELT Computing Eigenvectors of Linear Transformations……Page 639
READ Reading Questions……Page 646
EXC Exercises……Page 648
SOL Solutions……Page 649
TM Triangular Matrices……Page 653
UTMR Upper Triangular Matrix Representation……Page 654
OD Orthonormal Diagonalization……Page 658
NLT Nilpotent Linear Transformations……Page 663
PNLT Properties of Nilpotent Linear Transformations……Page 668
CFNLT Canonical Form for Nilpotent Linear Transformations……Page 672
IS Invariant Subspaces……Page 681
GEE Generalized Eigenvectors and Eigenspaces……Page 684
RLT Restrictions of Linear Transformations……Page 688
GESD Generalized Eigenspace Decomposition……Page 699
JCF Jordan Canonical Form……Page 705
CHT Cayley-Hamilton Theorem……Page 717
R Representations……Page 719
LS.MMA Linear Solve……Page 721
NS.MMA Null Space……Page 722
VFSS.MMA Vector Form of Solution Set……Page 723
GSP.MMA Gram-Schmidt Procedure……Page 724
ME.TI86 Matrix Entry……Page 725
TM.TI86 Transpose of a Matrix……Page 726
VLC.TI83 Vector Linear Combinations……Page 727
CNA Arithmetic with complex numbers……Page 729
CCN Conjugates of Complex Numbers……Page 731
MCN Modulus of a Complex Number……Page 732
SET Sets……Page 733
SC Set Cardinality……Page 734
SO Set Operations……Page 735
D Definitions……Page 737
L Language……Page 738
C Constructive Proofs……Page 739
N Negation……Page 740
CD Contradiction……Page 741
ME Multiple Equivalences……Page 742
DC Decompositions……Page 743
I Induction……Page 744
LC Lemmas and Corollaries……Page 745
Appendix A Archetypes……Page 749
A……Page 753
B……Page 757
C……Page 762
D……Page 766
E……Page 770
F……Page 773
G……Page 778
H……Page 782
I……Page 786
J……Page 790
K……Page 795
L……Page 799
M……Page 802
N……Page 804
O……Page 806
P……Page 809
Q……Page 811
R……Page 814
S……Page 817
T……Page 819
U……Page 821
V……Page 823
W……Page 825
X……Page 827
ADDENDUM: How to use this License for your documents……Page 831
F Fields……Page 839
FF Finite Fields……Page 840
EXC Exercises……Page 844
SOL Solutions……Page 846
T Trace……Page 847
EXC Exercises……Page 851
SOL Solutions……Page 852
HP Hadamard Product……Page 853
DMHP Diagonal Matrices and the Hadamard Product……Page 855
EXC Exercises……Page 858
VM Vandermonde Matrix……Page 859
PSM Positive Semi-Definite Matrices……Page 863
EXC Exercises……Page 866
ROD Rank One Decomposition……Page 867
TD Triangular Decomposition……Page 873
TDSSE Triangular Decomposition and Solving Systems of Equations……Page 875
CTD Computing Triangular Decompositions……Page 877
MAP Matrix-Adjoint Product……Page 881
SVD Singular Value Decomposition……Page 884
SRM Square Root of a Matrix……Page 887
POD Polar Decomposition……Page 891
CF Curve Fitting……Page 895
DF Data Fitting……Page 896
EXC Exercises……Page 898
SAS Sharing A Secret……Page 899
Reviews
There are no reviews yet.