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Linear Algebra I [Mīkstie vāki]

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  • Formāts: Paperback / softback, 261 pages, height x width: 254x178 mm, weight: 477 g
  • Sērija : Courant Lecture Notes
  • Izdošanas datums: 25-Jan-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470448718
  • ISBN-13: 9781470448714
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Linear Algebra IPalielināt
  • Formāts: Paperback / softback, 261 pages, height x width: 254x178 mm, weight: 477 g
  • Sērija : Courant Lecture Notes
  • Izdošanas datums: 25-Jan-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470448718
  • ISBN-13: 9781470448714
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470448714.html
Keywords:
The first of two volumes on linear algebra for graduate students and advanced undergraduates, this book explains the structure of linear operators as the key to solving problems in which they arise. It describes basic ideas of linear algebra, including vector spaces, linear operators, duality, determinants, diagonalization, and inner product spaces, with exercises and examples and an emphasis on proofs. Readers should have a prior undergraduate course in linear algebra, a basic understanding of matrix algebra, and some proficiency with mathematical proofs. Annotation ©2019 Ringgold, Inc., Portland, OR (protoview.com)
Preface ix
Organization of the Text x
Acknowledgments x
Chapter 1 Vector Spaces over a Field IK
1(48)
1.1 Vector Spaces
1(6)
Basic Properties and Examples
1(3)
Multivariate Polynomials and Multi-Index Notation
4(3)
Row Space and Column Space of a Matrix
7(1)
1.2 Vector Subspaces
7(5)
Systems of Linear Equations and Their Solutions
8(3)
M(n, R) and Other Spaces of Matrices
11(1)
1.3 Solving Matrix Equation Ax = b
12(9)
Elementary Operations and Echelon Form of Ax = b
13(3)
The Homogeneous Equation Ax = 0
16(1)
Solving Inhomogeneous Equations Ax = b
17(1)
Determining the Linear Span of a Set of Vectors
18(1)
Implicit and Parametric Descriptions of Vector Subspaces
19(1)
More on Elementary Row and Column Operations
20(1)
1.4 Linear Span, Independence, and Bases
21(10)
Existence and Construction of Bases
21(3)
The Dimension dim(K) of a Vector Space
24(2)
Implicit and Parametric Description of Subspaces (Revisited)
26(3)
The Lagrange Interpolation Formula
29(1)
Rank of a Matrix: Row Rank vs. Column Rank
30(1)
1.5 Quotient Spaces V/W
31(18)
Algebraic Structure in V/W
32(2)
Finding Bases in V/W and the Dimension Formula
34(4)
Additional Exercises
38(8)
Appendix: The Degree Formula for K[ x1, ..., xN]
46(3)
Chapter 2 Linear Operators T: V → W
49(40)
2.1 Definitions and General Facts
49(7)
Dimension Theorems for Kernel K(T) and Range R(T)
51(2)
Computing K(T) and R(T)
53(3)
2.2 Isomorphisms and Invariant Subspaces
56(6)
Eigenvalues, Eigenspaces, and the Characteristic Polynomial PT(x)
56(1)
Decomposition of Operators
57(2)
Isomorphisms of Vector Spaces
59(1)
More on Quotient Spaces
60(1)
First and Second Isomorphism Theorems for Quotient Spaces
60(2)
2.3 Direct Sums of Vector Spaces
62(7)
Projection Operators and Direct Sums
64(3)
Direct Sums and Diagonalization
67(1)
Independence of the Eigenspaces
67(2)
2.4 Representing Linear Operators as Matrices
69(20)
The Associated Matrices [ T]
69(3)
The Correspondence Between Matrices and Linear Operators
72(4)
Change of Basis and Similarity Transformations
76(3)
Similarity Classes in Matrix Space
79(1)
Similarity and RST Equivalence Relations
80(2)
Additional Exercises
82(7)
Chapter 3 Duality and the Dual Space V*
89(20)
3.1 Definitions and Examples
89(5)
Lengths and Orthogonality of Vectors in Inner Product Spaces
90(3)
Duality and the Fourier Transform
93(1)
3.2 Dual Bases in the Dual Space V*
94(3)
3.3 The Transpose TT: W* → V* of T: V → W
97(12)
Matrix Description of a Transpose TT
98(1)
Calculating the Transpose of a Projection Operator
99(2)
Reflexivity of Finite-Dimensional Spaces
101(2)
The Annihilator M° of a Subspace M in V
103(1)
A Dimension Formula for Annihilators M°
103(1)
Row Rank vs. Column Rank (Revisited)
104(1)
Outline of a Proof That (RowRank) = (ColRank)
104(1)
Additional Exercises
105(4)
Chapter 4 Determinants
109(30)
4.1 The Permutation Group Sn
109(9)
Cycles and the Cycle Decomposition Theorem
110(4)
Parity of a Permutation
114(3)
(Optional) An Alternative Proof of the Parity Theorem 4.17
117(1)
4.2 Determinants
118(21)
Proving the Basic Properties of det(A)
119(3)
Row Operations, Determinants, and Matrix Inverses
122(2)
Computing Matrix Inverses
124(2)
Computational Issues
126(1)
Proving the Multiplicative Property det(AB) = det(A)det(B)
127(2)
Defining Determinants of Linear Operators
129(1)
More on Rank, RowRank, and ColumnRank
130(2)
Expansion by Minors and Cramer's Rule
132(2)
Additional Exercises
134(5)
Chapter 5 The Diagonalization Problem
139(42)
5.1 Eigenvalues, Characteristic Polynomial, and Spectrum
139(6)
Factoring Polynomials
141(1)
Fundamental Theorem of Algebra
142(2)
The Quadratic Formula
144(1)
5.2 Eigenvalues and the Characteristic Polynomial
145(8)
Finding the Eigenspaces of T: V → V
147(3)
Example: Rotation Matrices in R2
150(2)
The Case of Distinct Eigenvalues
152(1)
5.3 Diagonalization and Limits of Operators
153(7)
Norms on Finite-Dimensional Spaces
153(1)
Multiplicative Properties of Norms on Matrix Space
154(1)
The Operator Norm ||T||op on Linear Operators and Matrices
155(2)
Equivalence of Norms on Finite-Dimensional Spaces
157(1)
Practical Calculations with Matrix Norms
158(1)
Convergence of Sequences and Series in Matrix Space
158(1)
Properties of Matrix Norms
159(1)
5.4 Application: Computing the Exponential eA of a Matrix
160(5)
The Cauchy Convergence Criterion in Matrix Space
161(1)
Convergence of the Exponential Series
162(1)
Explicit Computation of eA
162(3)
5.5 Application: Linear Systems of Differential Equations
165(4)
Example: Solving dx/dt = Ax
167(2)
5.6 Application: Matrix-Valued Geometric Series
169(12)
Convergence of Matrix-Valued Power Series
170(1)
Small Perturbations of Invertible Matrices
171(1)
Geometric Series for Nilpotent N
172(1)
Additional Exercises
172(9)
Chapter 6 Inner Product Spaces
181(76)
6.1 Basic Definitions and Examples
181(9)
Euclidean Norms on Rn and Cn
182(2)
The Cauchy-Schwarz Inequality
184(2)
Hilbert-Schmidt Norm
186(1)
Polarization Identity
187(1)
Orthonormal Bases in Inner Product Spaces
187(1)
Bessel's Inequality
188(2)
6.2 Orthogonal Complements and Projections
190(10)
Orthogonal Projections on Inner Product Spaces
190(2)
The Gram-Schmidt Construction
192(3)
The Legendre Polynomials
195(1)
Fourier Series Expansions
196(3)
A Geometry Problem
199(1)
6.3 Adjoints and Orthonormal Decompositions
200(9)
Diagonalization vs. Orthogonal Diagonalization
200(1)
Dual Spaces of Inner Product Spaces
201(1)
The Adjoint Operator T*: W → V
202(2)
Linear Projections vs. Orthogonal Projections
204(2)
Adjoint T* vs. Transpose TT
206(1)
Computing an Operator Adjoint
206(1)
Self-Adjoint, Unitary, and Normal Operators
207(2)
6.4 Diagonalization in Inner Product Spaces
209(13)
Orthogonal Diagonalization
209(1)
Schur Normal Form
209(3)
Diagonalizing Self-Adjoint and Normal Operators
212(4)
Unitary Equivalence of Operators vs. Similarity
216(3)
The Matrix Groups U(n), SU(n), O(n), SO(n)
219(2)
Change of Orthonormal Basis
221(1)
Diagonalization over K = C: A Summary
222(1)
6.5 Reflections, Rotations, and Rigid Motions on Rn
222(7)
The Group of Rigid Motions M(n)
223(1)
Reflections on Inner Product Spaces
224(2)
Euler's Theorem: Rotations on [ R3
226(2)
Further Comments on Euler
228(1)
6.6 Spectral Theorem for Vector and Inner Product Spaces
229(10)
Spectral Theorem
230(4)
Functions of Operators: eT and T Revisited
234(2)
Computing a Spectral Decomposition
236(1)
Determining the Spectral Projections Fx
237(2)
6.7 Positive Operators and Polar Decomposition
239(18)
Positive Square Roots
240(1)
Polar Decompositions T = UP (Invertible T)
241(2)
Computing the Polar Decomposition for Invertible T: V → W
243(1)
The Singular Value Decomposition
244(1)
Computing a Singular Value Decomposition
245(1)
The General Polar Decomposition
246(2)
Additional Exercises
248(9)
Index 257
Frederick P. Greenleaf, Courant Institute, New York University, NY.

Sophie Marques, University of Cape Town, South Africa.