Linear algebra as an introduction to abstract mathematics:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2016]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | x, 198 Seiten Illustrationen 25 cm |
| ISBN: | 9789814730358 9789814723770 |
Internformat
MARC
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| 100 | 1 | |a Lankham, Isaiah |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear algebra as an introduction to abstract mathematics |c Isaiah Lankham (California State University, East Bay, USA), Bruno Nachtergaele (University of California, Davis, USA), Anne Schilling (University of California, Davis, USA) |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2016] | |
| 300 | |a x, 198 Seiten |b Illustrationen |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Algebras, Linear | |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
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| 700 | 1 | |a Nachtergaele, Bruno |e Verfasser |4 aut | |
| 700 | 1 | |a Schilling, Anne |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028945542&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804176181007745024 |
|---|---|
| adam_text | Contents
Preface v
1. What is Linear Algebra? 1
1.1 Introduction...................................................... 1
1.2 What is Linear Algebra?........................................... 1
1.3 Systems of linear equations....................................... 3
1.3.1 Linear equations.......................................... 3
1.3.2 Non-linear equations...................................... 4
1.3.3 Linear transformations.................................... 5
1.3.4 Applications of linear equations.......................... 6
Exercises............................................................... 7
2. Introduction to Complex Numbers 9
2.1 Definition of complex numbers .................................... 9
2.2 Operations on complex numbers . ................................. 10
2.2.1 Addition and subtraction of complex numbers.............. 10
2.2.2 Multiplication and division of complex numbers........... 11
2.2.3 Complex conjugation...................................... 12
2.2.4 The modulus (a.k.a. norm., length, or magnitude) ........ 13
2.2.5 Complex numbers as vectors in M2......................... 14
2.3 Polar form and geometric interpretation for C.................... 15
2.3.1 Polar form for complex numbers .......................... 15
2.3.2 Geometric multiplication for complex numbers............. 16
2.3.3 Exponentiation and root extraction....................... 17
2.3.4 Some complex elementary functions........................ 18
Exercises.............................................................. 18
3. The Fundamental Theorem of Algebra and Factoring Polynomials 21
3.1 The Fundamental Theorem of Algebra............................... 21
3.2 Factoring polynomials............................................ 24
vii
vili Linear Algebra: As an Introduction to Abstract Mathematics
Exercises.............................................................. 26
4. Vector Spaces 29
4.1 Definition of vector spaces...................................... 29
4.2 Elementary properties of vector spaces........................... 31
4.3 Subspaces........................................................ 32
4.4 Sums and direct sums............................................. 34
Exercises.............................................................. 37
5. Span and Bases 39
5.1 Linear span...................................................... 39
5.2 Linear independence.............................................. 40
5.3 Bases............................................................ 44
5.4 Dimension........................................................ 46
Exercises.............................................................. 49
6. Linear Maps 51
6.1 Definition and elementary properties............................. 51
6.2 Null spaces...................................................... 53
6.3 Range............................................................ 55
6.4 Homomorphisms.................................................... 56
6.5 The dimension formula............................................ 56
6.6 The matrix of a linear map ...................................... 57
6.7 Invertibility.................................................... 61
Exercises.............................................................. 64
7. Eigenvalues and Eigenvectors 67
7.1 Invariant subspaces.............................................. 67
7.2 Eigenvalues...................................................... 68
7.3 Diagonal matrices................................................ 70
7.4 Existence of eigenvalues......................................... 71
7.5 Upper triangular matrices........................................ 72
7.6 Diagonalization of 2 x 2 matrices and applications .............. 75
Exercises.............................................................. 76
8. Permutations and the Determinant of a Square Matrix 81
8.1 Permutations..................................................... 81
8.1.1 Definition of permutations................................ 81
8.1.2 Composition of permutations............................... 84
8.1.3 Inversions and the sign of a permutation.................. 85
8.2 Determinants..................................................... 87
8.2.1 Summations indexed by the set of all permutations .... 87
Contents ix
8.2.2 Properties of the determinant ............................. 88
8.2.3 Further properties and applications........................ 91
8.2.4 Computing determinants with cofactor expansions............ 92
Exercises............................................................... 93
9. Inner Product Spaces 95
9.1 Inner product...................................................... 95
9.2 Norms.............................................................. 96
9.3 Orthogonality...................................................... 98
9.4 Orthonormal bases..................................................100
9.5 The Gram-Schmidt orthogonalization procedure.......................102
9.6 Orthogonal projections and minimization problems...................104
Exercises...............................................................107
10. Change of Bases 111
10.1 Coordinate vectors.................................................Ill
10.2 Change of basis transformation.....................................112
Exercises...............................................................115
11. The Spectral Theorem for Normal Linear Maps 117
11.1 Self-adjoint or hermitian operators................................117
11.2 Normal operators...................................................119
11.3 Normal operators and the spectral decomposition....................120
11.4 Applications of the Spectral Theorem: diagonalization..............121
11.5 Positive operators.................................................124
11.6 Polar decomposition................................................125
11.7 Singular-value decomposition.......................................126
Exercises...............................................................127
Appendices 129
Appendix A Supplementary Notes on Matrices and Linear Systems 129
A.l From linear systems to matrix equations............................129
A. 1.1 Definition of and notation for matrices ..................130
A. 1.2 Using matrices to encode linear systems.....................132
A.2 Matrix arithmetic..................................................134
A.2.1 Addition and scalar multiplication...........................134
A.2.2 Multiplication of matrices...................................137
A.2.3 Invertibility of square matrices.............................140
A.3 Solving linear systems by factoring the coefficient matrix.........142
A.3.1 Factorizing matrices using Gaussian elimination..............142
A.3.2 Solving homogeneous linear systems...........................149
щ
х Linear Algebra: As an Introduction to Abstract Mathematics
A.3.3 Solving inhomogeneous linear systems........................152
A.3.4 Solving linear systems with LU-factorization...............155
A.4 Matrices and linear maps .........................................159
A.4.1 The canonical matrix of a linear map.......................159
A.4.2 Using linear maps to solve linear systems..................160
A. 5 Special operations on matrices....................................164
A.5.1 Transpose and conjugate transpose .........................164
A.5.2 The trace of a square matrix...............................165
Exercises...............................................................166
Appendix В The Language of Sets and Functions 171
B. l Sets............................................................ 171
B.2 Subset, union, intersection, and Cartesian product................173
B.3 Relations ........................................................174
B. 4 Functions.........................................................175
Appendix C Summary of Algebraic Structures Encountered 177
C. l Binary operations and scaling operations..........................177
C.2 Groups, fields, and vector spaces............................... 180
C.3 Rings and algebras................................................183
Appendix D Some Common Math Symbols and Abbreviations 185
Index 191
|
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| language | English |
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| spelling | Lankham, Isaiah Verfasser aut Linear algebra as an introduction to abstract mathematics Isaiah Lankham (California State University, East Bay, USA), Bruno Nachtergaele (University of California, Davis, USA), Anne Schilling (University of California, Davis, USA) New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2016] x, 198 Seiten Illustrationen 25 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s DE-604 Nachtergaele, Bruno Verfasser aut Schilling, Anne Verfasser aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028945542&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lankham, Isaiah Nachtergaele, Bruno Schilling, Anne Linear algebra as an introduction to abstract mathematics Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Linear algebra as an introduction to abstract mathematics |
| title_auth | Linear algebra as an introduction to abstract mathematics |
| title_exact_search | Linear algebra as an introduction to abstract mathematics |
| title_full | Linear algebra as an introduction to abstract mathematics Isaiah Lankham (California State University, East Bay, USA), Bruno Nachtergaele (University of California, Davis, USA), Anne Schilling (University of California, Davis, USA) |
| title_fullStr | Linear algebra as an introduction to abstract mathematics Isaiah Lankham (California State University, East Bay, USA), Bruno Nachtergaele (University of California, Davis, USA), Anne Schilling (University of California, Davis, USA) |
| title_full_unstemmed | Linear algebra as an introduction to abstract mathematics Isaiah Lankham (California State University, East Bay, USA), Bruno Nachtergaele (University of California, Davis, USA), Anne Schilling (University of California, Davis, USA) |
| title_short | Linear algebra as an introduction to abstract mathematics |
| title_sort | linear algebra as an introduction to abstract mathematics |
| topic | Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algebras, Linear Lineare Algebra |
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