book

Linear Algebra, 5th Edition

Name: Linear Algebra, 5th Edition
ISBN: 9780136745495

by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

September 2018

Intermediate to advanced

580 pages

16h 3m

English

Pearson

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Linear Algebra
Contents
Preface
Suggested Course OutlinesOverview of ContentsDifferences Between the Fourth and Fifth Editions
To the Student
1 Vector Spaces
1.1 Introduction
Exercises
1.2 Vector Spaces
Exercises
1.3 Subspaces
ExercisesDefinition.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 ConsequencesThe Dimension of SubspacesCorollary.The Lagrange Interpolation FormulaExercises
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
ExercisesDefinitions.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 MatrixDefinition.Exercises
3.3 Systems of Linear Equations—Theoretical Aspects
An ApplicationExercises
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
ExercisesDefinition.
4.4 Summary—Important Facts about Determinants
Properties of the DeterminantExercises
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 DiagonalizabilitySystems of Differential EquationsDirect Sums*Definition.Definition.Proof.Proof.ExercisesDefinitions.
5.3* Matrix Limits and Markov Chains
Applications*ExercisesDefinition.
5.4 Invariant Subspaces and the Cayley-Hamilton Theorem
The Cayley-Hamilton TheoremProof.Corollary (Cayley-Hamilton Theorem for Matrices).Proof.Invariant Subspaces and Direct Sums3Proof.Definition.Proof.ExercisesDefinition.
Index of Definitions for Chapter 5
6 Inner Product Spaces
6.1 Inner Products and Norms
ExercisesDefinition.
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 ApproximationLemma 1.Proof.Lemma 2.Proof.Corollary.Minimal Solutions to Systems of Linear EquationsProof.ExercisesDefinition.
6.4 Normal and Self-Adjoint Operators
ExercisesDefinitions.
6.5 Unitary and Orthogonal Operators and Their Matrices
Rigid Motions*Definition.Proof.Orthogonal Operators on R2Proof.Corollary.Conic SectionsExercisesDefinition.
6.6 Orthogonal Projections and the Spectral Theorem
Exercises
6.7* The Singular Value Decomposition and the Pseudoinverse
The Polar Decomposition of a Square MatrixProof.The PseudoinverseDefinition.The Pseudoinverse of a MatrixThe Pseudoinverse and Systems of Linear EquationsProof.Proof.Exercises
6.8* Bilinear and Quadratic Forms
Bilinear FormsDefinition.Definitions.Proof.Definition.Proof.Corollary 1.Proof.Corollary 2.Corollary 3.Definition.Proof.Corollary.Proof.Symmetric Bilinear FormsDefinition.Proof.Definition.Corollary.Proof.Proof.Proof.Corollary.Proof.Diagonalization of Symmetric MatricesQuadratic FormsDefinition.Quadratic Forms on a Real Inner Product SpaceProof.Corollary.Proof.The Second Derivative Test for Functions of Several VariablesProof.Sylvester’s Law of InertiaProof.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 ContractionExercises
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
ExercisesDefinitions.Definition.
7.3 The Minimal Polynomial
ExercisesDefinition.
7.4* The Rational Canonical Form
Uniqueness of the Rational Canonical FormLemma 1.Lemma 2.Corollary.Definitions.Direct Sums*Proof.Proof.Exercises
Index of Definitions for Chapter 7
Appendices
Appendix A Sets
Appendix B Functions
Appendix C Fields
Definitions.Proof.Corollary.Proof.Proof.Corollary.
Appendix D Complex Numbers
Definitions.Proof.Definition.Proof.Definition.Proof.Proof.Corollary.Proof.
Appendix E Polynomials
Definition.Proof.Corollary 1.Proof.Corollary 2.Proof.Definition.Lemma.Proof.Proof.Definitions.Proof.Proof.Proof.Definitions.Proof.Proof.Proof.Corollary.Proof.Proof.Proof.
Answers to Selected Exercises
Chapter 1
Section 1.1
Section 1.2
Section 1.3
Section 1.4
Section 1.5
Section 1.6
Section 1.7
Chapter 2
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Section 2.6
Section 2.7
Chapter 3
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Chapter 4
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Chapter 5
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Chapter 6
Section 6.1
Section 6.2
Section 6.3
Section 6.4
Section 6.5
Section 6.6
Section 6.7
Section 6.8
Section 6.9
Section 6.10
Section 6.11
Chapter 7
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Index
List of Symbols

Content preview from Linear Algebra, 5th Edition

3.2 The Rank of a Matrix and Matrix Inverses

In this section, we define the rank of a matrix. We then use elementary operations to compute the rank of a matrix and a linear transformation. The section concludes with a procedure for computing the inverse of an invertible matrix.

Definition.

If $A \in M_{m \times n} (F)$ , we define the rank of A, denoted rank(A), to be the rank of the linear transformation $L_{A} : F^{n} \to F^{m}$ .

Many results about the rank of a matrix follow immediately from the corresponding facts about a linear transformation. An important result of this type, which follows from Fact 3 (p. 101) and Corollary 2 to Theorem 2.18 (p. 103), is that an $n \times n$ matrix is invertible if and only if its rank is n.

Every matrix A is the matrix representation of the linear ...

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Linear Algebra, 5th Edition

by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

3.2 The Rank of a Matrix and Matrix Inverses

Definition.

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