Finite-dimensional linear algebra:
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2010
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| Series: | Discrete mathematics and its applications
59 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 650 S. graph. Darst. |
| ISBN: | 9781439815632 1439815631 |
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MARC
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| 100 | 1 | |a Gockenbach, Mark S. |d 1963- |e Verfasser |0 (DE-588)138322090 |4 aut | |
| 245 | 1 | 0 | |a Finite-dimensional linear algebra |c Mark S. Gockenbach |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press |c 2010 | |
| 300 | |a XXI, 650 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Discrete mathematics and its applications |v 59 | |
| 650 | 0 | |a Algebras, Linear | |
| 650 | 0 | |a Dimensional analysis | |
| 650 | 0 | |a Finite fields (Algebra) | |
| 650 | 0 | |a Vector spaces | |
| 650 | 0 | 7 | |a Dimension n |0 (DE-588)4309313-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Discrete mathematics and its applications |v 59 |w (DE-604)BV023551867 |9 59 | |
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Record in the Search Index
| _version_ | 1804143215089025024 |
|---|---|
| adam_text | Contents
Preface
xv
About the author
xxi
1
Some problems posed on vector spaces
1
1.1
Linear equations
......................... 1
1.1.1
Systems of linear algebraic equations
.......... 1
1.1.2
Linear ordinary differential equations
.......... 4
1.1.3
Some interpretation: The structure of the solution set
to a linear equation
................... 5
1.1.4
Finite fields and applications in discrete mathematics
7
1.2
Best approximation
....................... 8
1.2.1
Overdetermined linear systems
............. 8
1.2.2
Best approximation by a polynomial
.......... 11
1.3
Diagonalization
......................... 13
1.4
Summary
............................. 17
2
Fields and vector spaces
19
2.1
Fields
............................... 19
2.1.1
Definition and examples
................. 19
2.1.2
Basic properties of fields
................. 21
2.2
Vector spaces
........................... 29
2.2.1
Examples of vector spaces
................ 31
2.3
Subspaces
............................. 38
2.4
Linear combinations and spanning sets
............ 43
2.5
Linear independence
....................... 50
2.6
Basis and dimension
....................... 57
2.7
Properties of bases
........................ 66
2.8
Polynomial interpolation and the
Lagrange
basis
....... 73
2.8.1
Secret sharing
....................... 77
2.9
Continuous piecewise polynomial functions
.......... 82
2.9.1
Continuous piecewise linear functions
......... 84
2.9.2
Continuous piecewise quadratic functions
....... 87
2.9.3
Error in polynomial interpolation
............ 90
ix
Contents
Linear Operators 93
3.1 Linear
operators
......................... 93
3.1.1 Matrix Operators..................... 95
3.2
More properties of
linear
operators
............... 101
3.2.1
Vector spaces of operators
................ 101
3.2.2
The matrix of a linear operator on Euclidean spaces
. 101
3.2.3
Derivative and differential operators
.......... 103
3.2.4
Representing spanning sets and bases using matrices
. 103
3.2.5
The transpose of a matrix
................ 104
3.3
Isomorphic vector spaces
.................... 107
3.3.1
Injective and surjective functions; inverses
....... 108
3.3.2
The matrix of a linear operator on general vector spaces 111
3.4
Linear operator equations
.................... 116
3.4.1
Homogeneous linear equations
............. 117
3.4.2
Inhomogeneous linear equations
............. 118
3.4.3
General solutions
..................... 120
3.5
Existence and uniqueness of solutions
............. 124
3.5.1
The kernel of a linear operator and injectivity
..... 124
3.5.2
The rank of a linear operator and surjectivity
..... 126
3.5.3
Existence and uniqueness
................ 128
3.6
The fundamental theorem; inverse operators
......... 131
3.6.1
The inverse of a linear operator
............. 133
3.6.2
The inverse of a matrix
................. 134
3.7
Gaussian elimination
...................... 142
3.7.1
Computing A^1
..................... 148
3.7.2
Fields other than
R
................... 149
3.8
Newton s method
........................ 153
3.9
Linear ordinary differential equations
............. 158
3.9.1
The dimension of ker(L)
................. 158
3.9.2
Finding a basis for ker(L)
................ 161
3.9.2.1
The easy case: Distinct real roots
...... 162
3.9.2.2
The case of repeated real roots
........ 162
3.9.2.3
The case of complex roots
........... 163
3.9.3
The Wronskian test for linear independence
...... 163
3.9.4
The Vandermonde matrix
................ 166
3.10
Graph theory
........................... 168
3.10.1
The incidence matrix of a graph
............ 168
3.10.2
Walks and matrix multiplication
............ 169
3.10.3
Graph isomorphisms
................... 171
3.11
Coding theory
.......................... 175
3.11.1
Generator matrices; encoding and decoding
...... 177
3.11.2
Error correction
..................... 179
3.11.3
The probability of errors
................. 181
3.12
Linear programming
....................... 183
3.12.1
Specification of linear programming problems
..... 184
Contents
Xl
3.12.2 Basic
theory .......................
186
3.12.3
The simplex method
................... 191
3.12.3.1
Finding an initial BFS
............ 196
3.12.3.2
Unbounded LPs
................ 199
3.12.3.3
Degeneracy and cycling
............ 200
3.12.4
Variations on the standard LPs
............. 202
Determinants and eigenvalues
205
4.1
The determinant function
.................... 206
4.1.1
Permutations
....................... 210
4.1.2
The complete expansion of the determinant
...... 212
4.2
Further properties of the determinant function
........ 217
4.3
Practical computation of
det
(A)
................ 221
4.3.1
A recursive formula for dct(A)
............. 224
4.3.2
Cramer s rule
....................... 226
4.4
A note about polynomials
.................... 230
4.5
Eigenvalues and the characteristic polynomial
........ 232
4.5.1
Eigenvalues of real matrix
................ 235
4.6
Diagonalization
......................... 241
4.7
Eigenvalues of linear operators
................. 251
4.8
Systems of linear ODEs
..................... 257
4.8.1
Complex eigenvalues
................... 259
4.8.2
Solving the initial value problem
............ 260
4.8.3
Linear systems in matrix form
............. 261
4.9
Integer programming
...................... 265
4.9.1
Totally unimodular matrices
.............. 265
4.9.2
Transportation problems
................. 268
The Jordan canonical form
273
5.1
Invariant subspaces
....................... 273
5.1.1
Direct sums
........................ 276
5.1.2 Eigenspaces
and generalized eigenspaces
........ 277
5.2
Generalized eigenspaces
..................... 283
5.2.1
Appendix: Beyond generalized eigenspaces
...... 290
5.2.2
The Cayley-Hamilton theorem
............. 294
5.3 Nilpotent
operators
....................... 300
5.4
The Jordan canonical form of a matrix
............ 309
5.5
The matrix exponential
..................... 318
5.5.1
Definition of the matrix exponential
.......... 319
5.5.2
Computing the matrix exponential
........... 319
5.6
Graphs and eigenvalues
..................... 325
5.6.1
Cospectral graphs
.................... 325
5.6.2
Bipartite graphs and eigenvalues
............ 326
5.6.3
Regular graphs
...................... 328
5.6.4
Distinct eigenvalues of a graph
............. 330
xii Contents
6
Orthogonality and best approximation
333
6.1
Norms and inner products
................... 333
6.1.1
Examples of norms and inner products
......... 337
6.2
The adjoint of a linear operator
................ 342
6.2.1
The adjoint of a linear operator
............. 343
6.3
Orthogonal vectors and bases
.................. 350
6.3.1
Orthogonal bases
..................... 351
6.4
The projection theorem
..................... 357
6.4.1
Overdetermined linear systems
............. 361
6.5
The Gram-Schmidt process
................... 368
6.5.1
Least-squares polynomial approximation
........ 371
6.6
Orthogonal complements
.................... 377
6.6.1
The fundamental theorem of linear algebra revisited
. 381
6.7
Complex inner product spaces
................. 386
6.7.1
Examples of complex inner product spaces
....... 388
6.7.2
Orthogonality in complex inner product spaces
.... 389
6.7.3
The adjoint of a linear operator
............. 390
6.8
More on polynomial approximation
.............. 394
6.8.1
A weighted L2 inner product
.............. 397
6.9
The energy inner product and Galerkin s method
...... 401
6.9.1
Piecewise polynomials
.................. 404
6.9.2
Continuous piecewise quadratic functions
....... 407
6.9.3
Higher degree finite element spaces
........... 409
6.10
Gaussian quadrature
...................... 411
6.10.1
The trapezoidal rule and Simpson s rule
........ 412
6.10.2
Gaussian quadrature
................... 413
6.10.3
Orthogonal polynomials
................. 415
6.10.4
Weighted Gaussian quadrature
............. 419
6.11
The Helmholtz decomposition
................. 420
6.11.1
The divergence theorem
................. 421
6.11.2
Stokes s theorem
..................... 422
6.11.3
The Helmholtz decomposition
.............. 423
7
The spectral theory of symmetric matrices
425
7.1
The spectral theorem for symmetric matrices
......... 425
7.1.1
Symmetric positive definite matrices
.......... 428
7.1.2
Hermitian matrices
.................... 430
7.2
The spectral theorem for normal matrices
........... 434
7.2.1
Outer products and the spectral decomposition
.... 437
7.3
Optimization and the Hessian matrix
............. 440
7.3.1
Background
........................ 440
7.3.2
Optimization of quadratic functions
.......... 441
7.3.3
Taylor s theorem
..................... 443
7.3.4
First- and second-order optimality conditions
..... 444
7.3.5
Local quadratic approximations
............. 446
Contents xiii
7.4 Lagrange
multipliers.......................
448
7.5
Spectral methods for differential equations
.......... 453
7.5.1
Eigenpairs of the differential operator
......... 454
7.5.2
Solving the BVP using eigenfunctions
......... 456
The singular value decomposition
463
8.1
Introduction to the
SVD
.................... 463
8.1.1
The
SVD
for singular matrices
............. 467
8.2
The
SVD
for general matrices
................. 470
8.3
Solving least-squares problems using the
SVD
........ 476
8.4
The
SVD
and linear inverse problems
............. 483
8.4.1
Resolving inverse problems through regularization
. . 489
8.4.2
The truncated
SVD
method
............... 489
8.4.3
Tikhonov regularization
................. 490
8.5
The Smith normal form of a matrix
.............. 494
8.5.1
An algorithm to compute the Smith normal form
. . . 495
8.5.2
Applications of the Smith normal form
......... 501
Matrix factorizations and numerical linear algebra
507
9.1
The
LU
factorization
...................... 507
9.1.1
Operation counts
..................... 512
9.1.2
Solving Ax
=
b
using the
LU
factorization
....... 514
9.2
Partial pivoting
......................... 516
9.2.1
Finite-precision arithmetic
................ 517
9.2.2
Examples of errors in Gaussian elimination
...... 518
9.2.3
Partial pivoting
...................... 519
9.2.4
The
PLU
factorization
.................. 522
9.3
The Cholesky factorization
................... 524
9.4
Matrix norms
.......................... 530
9.4.1
Examples of induced matrix norms
........... 534
9.5
The sensitivity of linear systems to errors
........... 537
9.6
Numerical stability
....................... 542
9.6.1
Backward error analysis
................. 543
9.6.2
Analysis of Gaussian elimination with partial pivoting
545
9.7
The sensitivity of the least-squares problem
.......... 548
9.8
The QR factorization
...................... 554
9.8.1
Solving the least-squares problem
............ 556
9.8.2
Computing the QR factorization
............ 556
9.8.3
Backward stability of the Householder QR algorithm
. 561
9.8.4
Solving a linear system
................. 562
9.9
Eigenvalues and simultaneous iteration
............ 564
9.9.1
Reduction to triangular form
.............. 564
9.9.2
The power method
.................... 566
9.9.3
Simultaneous iteration
.................. 567
9.10
The QR algorithm
........................ 572
xiv Contents
9.10.1
A practical QR algorithm
................ 573
9.10.1.1
Reduction to upper
Hessenberg
form
..... 574
9.10.1.2
The explicitly shifted QR algorithm
..... 576
9.10.1.3
The implicitly shifted QR algorithm
..... 579
10
Analysis in vector spaces
581
10.1
Analysis in R
.......................... 581
10.1.1
Convergence and continuity in Rn
........... 582
10.1.2
Compactness
....................... 584
10.1.3
Completeness of R
................... 586
10.1.4
Equivalence of norms on Rn
.............. 586
10.2
Infinite-dimensional vector spaces
............... 590
10.2.1
Banach and Hubert spaces
............... 592
10.3
Functional analysis
....................... 596
10.3.1
The dual of a Hilbert space
............... 600
10.4
Weak convergence
........................ 605
10.4.1
Convexity
......................... 611
A The Euclidean algorithm
617
A.
0.1
Computing multiplicative inverses in Zp
........ 618
A.0.2 Related results
...................... 619
В
Permutations
621
С
Polynomials
625
C.I Rings of polynomials
...................... 625
C.2 Polynomial functions
...................... 630
C.2.1 Factorization of polynomials
............... 632
D
Summary of analysis in
R
633
D.0.1 Convergence
....................... 633
D.0.2 Completeness of
R
.................... 634
D.
0.3
Open and closed sets
................... 635
D.O.4
Continuous functions
................... 636
Bibliography
637
Index
641
|
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| id | DE-604.BV036609972 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:44:07Z |
| institution | BVB |
| isbn | 9781439815632 1439815631 |
| language | English |
| lccn | 2010008253 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020530260 |
| oclc_num | 700637574 |
| open_access_boolean | |
| owner | DE-634 DE-703 DE-824 |
| owner_facet | DE-634 DE-703 DE-824 |
| physical | XXI, 650 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | CRC Press |
| record_format | marc |
| series | Discrete mathematics and its applications |
| series2 | Discrete mathematics and its applications |
| spelling | Gockenbach, Mark S. 1963- Verfasser (DE-588)138322090 aut Finite-dimensional linear algebra Mark S. Gockenbach Boca Raton, Fla. [u.a.] CRC Press 2010 XXI, 650 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications 59 Algebras, Linear Dimensional analysis Finite fields (Algebra) Vector spaces Dimension n (DE-588)4309313-9 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Dimension n (DE-588)4309313-9 s DE-604 Discrete mathematics and its applications 59 (DE-604)BV023551867 59 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020530260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gockenbach, Mark S. 1963- Finite-dimensional linear algebra Discrete mathematics and its applications Algebras, Linear Dimensional analysis Finite fields (Algebra) Vector spaces Dimension n (DE-588)4309313-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4309313-9 (DE-588)4035811-2 |
| title | Finite-dimensional linear algebra |
| title_auth | Finite-dimensional linear algebra |
| title_exact_search | Finite-dimensional linear algebra |
| title_full | Finite-dimensional linear algebra Mark S. Gockenbach |
| title_fullStr | Finite-dimensional linear algebra Mark S. Gockenbach |
| title_full_unstemmed | Finite-dimensional linear algebra Mark S. Gockenbach |
| title_short | Finite-dimensional linear algebra |
| title_sort | finite dimensional linear algebra |
| topic | Algebras, Linear Dimensional analysis Finite fields (Algebra) Vector spaces Dimension n (DE-588)4309313-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algebras, Linear Dimensional analysis Finite fields (Algebra) Vector spaces Dimension n Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020530260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023551867 |
| work_keys_str_mv | AT gockenbachmarks finitedimensionallinearalgebra |