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Linear algebra is a foundation course for students entering mathematics, engineering, and computer science, and the fourth edition includes more problems connected directly with applications to these majors. It is also updated throughout to include new essential appendices in algebraic systems, polynomials, and matrix applications.
Author(s): Seymour Lipschutz; Marc Lipson
Series: Schaum's Outlines
Edition: 4
Publisher: McGraw-Hill Education
Year: 2008
Language: English
Commentary: decrypted from 5EFD37C573617C67BC148A65EAC06129 source file
Pages: 480
Contents
Chapter 1 Vectors in R[sup(n)] and C[sup(n)], Spatial Vectors
1.1 Introduction
1.2 Vectors in R[sup(n)]
1.3 Vector Addition and Scalar Multiplication
1.4 Dot (Inner) Product
1.5 Located Vectors, Hyperplanes, Lines, Curves in R[sup(n)]
1.6 Vectors in R[sup(3)] (Spatial Vectors), ijk Notation
1.7 Complex Numbers
1.8 Vectors in C[sup(n)]
Chapter 2 Algebra of Matrices
2.1 Introduction
2.2 Matrices
2.3 Matrix Addition and Scalar Multiplication
2.4 Summation Symbol
2.5 Matrix Multiplication
2.6 Transpose of a Matrix
2.7 Square Matrices
2.8 Powers of Matrices, Polynomials in Matrices
2.9 Invertible (Nonsingular) Matrices
2.10 Special Types of Square Matrices
2.11 Complex Matrices
2.12 Block Matrices
Chapter 3 Systems of Linear Equations
3.1 Introduction
3.2 Basic Definitions, Solutions
3.3 Equivalent Systems, Elementary Operations
3.4 Small Square Systems of Linear Equations
3.5 Systems in Triangular and Echelon Forms
3.6 Gaussian Elimination
3.7 Echelon Matrices, Row Canonical Form, Row Equivalence
3.8 Gaussian Elimination, Matrix Formulation
3.9 Matrix Equation of a System of Linear Equations
3.10 Systems of Linear Equations and Linear Combinations of Vectors
3.11 Homogeneous Systems of Linear Equations
3.12 Elementary Matrices
3.13 LU Decomposition
Chapter 4 Vector Spaces
4.1 Introduction
4.2 Vector Spaces
4.3 Examples of Vector Spaces
4.4 Linear Combinations, Spanning Sets
4.5 Subspaces
4.6 Linear Spans, Row Space of a Matrix
4.7 Linear Dependence and Independence
4.8 Basis and Dimension
4.9 Application to Matrices, Rank of a Matrix
4.10 Sums and Direct Sums
4.11 Coordinates
Chapter 5 Linear Mappings
5.1 Introduction
5.2 Mappings, Functions
5.3 Linear Mappings (Linear Transformations)
5.4 Kernel and Image of a Linear Mapping
5.5 Singular and Nonsingular Linear Mappings, Isomorphisms
5.6 Operations with Linear Mappings
5.7 Algebra A(V) of Linear Operators
Chapter 6 Linear Mappings and Matrices
6.1 Introduction
6.2 Matrix Representation of a Linear Operator
6.3 Change of Basis
6.4 Similarity
6.5 Matrices and General Linear Mappings
Chapter 7 Inner Product Spaces, Orthogonality
7.1 Introduction
7.2 Inner Product Spaces
7.3 Examples of Inner Product Spaces
7.4 Cauchy–Schwarz Inequality, Applications
7.5 Orthogonality
7.6 Orthogonal Sets and Bases
7.7 Gram–Schmidt Orthogonalization Process
7.8 Orthogonal and Positive Definite Matrices
7.9 Complex Inner Product Spaces
7.10 Normed Vector Spaces (Optional)
Chapter 8 Determinants
8.1 Introduction
8.2 Determinants of Orders 1 and 2
8.3 Determinants of Order 3
8.4 Permutations
8.5 Determinants of Arbitrary Order
8.6 Properties of Determinants
8.7 Minors and Cofactors
8.8 Evaluation of Determinants
8.9 Classical Adjoint
8.10 Applications to Linear Equations, Cramer's Rule
8.11 Submatrices, Minors, Principal Minors
8.12 Block Matrices and Determinants
8.13 Determinants and Volume
8.14 Determinant of a Linear Operator
8.15 Multilinearity and Determinants
Chapter 9 Diagonalization: Eigenvalues and Eigenvectors
9.1 Introduction
9.2 Polynomials of Matrices
9.3 Characteristic Polynomial, Cayley–Hamilton Theorem
9.4 Diagonalization, Eigenvalues and Eigenvectors
9.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices
9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms
9.7 Minimal Polynomial
9.8 Characteristic and Minimal Polynomials of Block Matrices
Chapter 10 Canonical Forms
10.1 Introduction
10.2 Triangular Form
10.3 Invariance
10.4 Invariant Direct-Sum Decompositions
10.5 Primary Decomposition
10.6 Nilpotent Operators
10.7 Jordan Canonical Form
10.8 Cyclic Subspaces
10.9 Rational Canonical Form
10.10 Quotient Spaces
Chapter 11 Linear Functionals and the Dual Space
11.1 Introduction
11.2 Linear Functionals and the Dual Space
11.3 Dual Basis
11.4 Second Dual Space
11.5 Annihilators
11.6 Transpose of a Linear Mapping
Chapter 12 Bilinear, Quadratic, and Hermitian Forms
12.1 Introduction
12.2 Bilinear Forms
12.3 Bilinear Forms and Matrices
12.4 Alternating Bilinear Forms
12.5 Symmetric Bilinear Forms, Quadratic Forms
12.6 Real Symmetric Bilinear Forms, Law of Inertia
12.7 Hermitian Forms
Chapter 13 Linear Operators on Inner Product Spaces
13.1 Introduction
13.2 Adjoint Operators
13.3 Analogy Between A(V) and C, Special Linear Operators
13.4 Self-Adjoint Operators
13.5 Orthogonal and Unitary Operators
13.6 Orthogonal and Unitary Matrices
13.7 Change of Orthonormal Basis
13.8 Positive Definite and Positive Operators
13.9 Diagonalization and Canonical Forms in Inner Product Spaces
13.10 Spectral Theorem
Appendix A: Multilinear Products
Appendix B: Algebraic Structures
Appendix C: Polynomials over a Field
Appendix D: Odds and Ends
List of Symbols
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z
