Linear Algebra 5th Edition by Stephen Friedberg 0134860241 9780134860244

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ISBN 10: 0134860241
ISBN 13: 9780134860244
Author: Stephen H. Friedberg

Linear Algebra 5th Table of contents:

1 Vector Spaces
1.1 Introduction
Exercises
1.2 Vector Spaces
Exercises
1.3 Subspaces
Exercises
Definition.
Definition.
1.4 Linear Combinations and Systems of Linear Equations
Exercises
1.5 Linear Dependence and Linear Independence
Exercises
1.6 Bases and Dimension
An Overview of Dimension and Its Consequences
The Dimension of Subspaces
Corollary.
The Lagrange Interpolation Formula
Exercises
1.7* Maximal Linearly Independent Subsets
Exercises
Index of Definitions for Chapter 1
2 Linear Transformations and Matrices
2.1 Linear Transformations, Null Spaces, and Ranges
Exercises
Definitions.
Definitions.
2.2 The Matrix Representation of a Linear Transformation
Exercises
2.3 Composition of Linear Transformations and Matrix Multiplication
Applications*
Exercises
2.4 Invertibility and Isomorphisms
Exercises
2.5 The Change of Coordinate Matrix
Exercises
2.6* Dual Spaces
Exercises
2.7* Homogeneous Linear Differential Equations with Constant Coefficients
Exercises
Index of Definitions for Chapter 2
3 Elementary Matrix Operations and Systems of Linear Equations
3.1 Elementary Matrix Operations and Elementary Matrices
Exercises
3.2 The Rank of a Matrix and Matrix Inverses
The Inverse of a Matrix
Definition.
Exercises
3.3 Systems of Linear Equations—Theoretical Aspects
An Application
Exercises
3.4 Systems of Linear Equations—Computational Aspects
Exercises
Index of Definitions for Chapter 3
4 Determinants
4.1 Determinants of Order 2
Exercises
4.2 Determinants of Order n
Exercises
4.3 Properties of Determinants
Exercises
Definition.
4.4 Summary—Important Facts about Determinants
Properties of the Determinant
Exercises
4.5* A Characterization of the Determinant
Exercises
Index of Definitions for Chapter 4
5 Diagonalization
5.1 Eigenvalues and Eigenvectors
Exercises
5.2 Diagonalizability
Test for Diagonalizability
Systems of Differential Equations
Direct Sums*
Definition.
Definition.
Proof.
Proof.
Exercises
Definitions.
5.3* Matrix Limits and Markov Chains
Applications*
Exercises
Definition.
5.4 Invariant Subspaces and the Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem
Proof.
Corollary (Cayley-Hamilton Theorem for Matrices).
Proof.
Invariant Subspaces and Direct Sums3
Proof.
Definition.
Proof.
Exercises
Definition.
Index of Definitions for Chapter 5
6 Inner Product Spaces
6.1 Inner Products and Norms
Exercises
Definition.
6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
Exercises
6.3 The Adjoint of a Linear Operator
Proof.
Proof.
Proof.
Corollary.
Proof.
Proof.
Corollary.
Proof.
Least Squares Approximation
Lemma 1.
Proof.
Lemma 2.
Proof.
Corollary.
Minimal Solutions to Systems of Linear Equations
Proof.
Exercises
Definition.
6.4 Normal and Self-Adjoint Operators
Exercises
Definitions.
6.5 Unitary and Orthogonal Operators and Their Matrices
Rigid Motions*
Definition.
Proof.
Orthogonal Operators on R2
Proof.
Corollary.
Conic Sections
Exercises
Definition.
6.6 Orthogonal Projections and the Spectral Theorem
Exercises
6.7* The Singular Value Decomposition and the Pseudoinverse
The Polar Decomposition of a Square Matrix
Proof.
The Pseudoinverse
Definition.
The Pseudoinverse of a Matrix
The Pseudoinverse and Systems of Linear Equations
Proof.
Proof.
Exercises
6.8* Bilinear and Quadratic Forms
Bilinear Forms
Definition.
Definitions.
Proof.
Definition.
Proof.
Corollary 1.
Proof.
Corollary 2.
Corollary 3.
Definition.
Proof.
Corollary.
Proof.
Symmetric Bilinear Forms
Definition.
Proof.
Definition.
Corollary.
Proof.
Proof.
Proof.
Corollary.
Proof.
Diagonalization of Symmetric Matrices
Quadratic Forms
Definition.
Quadratic Forms on a Real Inner Product Space
Proof.
Corollary.
Proof.
The Second Derivative Test for Functions of Several Variables
Proof.
Sylvester’s Law of Inertia
Proof.
Definitions.
Corollary 1 (Sylvester’s Law of Inertia for Matrices).
Definitions.
Corollary 2.
Proof.
Corollary 3.
Exercises
6.9* Einstein’s Special Theory of Relativity
Time Contraction
Exercises
6.10* Conditioning and the Rayleigh Quotient
Exercises
6.11* The Geometry of Orthogonal Operators
Exercises
Index of Definitions for Chapter 6
7 Canonical Forms
7.1 The Jordan Canonical Form I
Exercises
7.2 The Jordan Canonical Form II
Exercises
Definitions.
Definition.
7.3 The Minimal Polynomial
Exercises
Definition.
7.4* The Rational Canonical Form
Uniqueness of the Rational Canonical Form
Lemma 1.
Lemma 2.
Corollary.
Definitions.
Direct Sums*
Proof.
Proof.

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