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# Linear Algebra

## Second EditionTed Shifrin; Malcolm Adams

©2011
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*Linear Algebra: A Geometric Approach, Second Edition*, presents the standard computational aspects of linear algebra and includes a variety of intriguing interesting applications that would be interesting to motivate science and engineering students, as well as help mathematics students make the transition to more abstract advanced courses. The text guides students on how to think about mathematical concepts and write rigorous mathematical arguments.

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Learn More## Table of Contents

Preface

Foreword to the Instructor

Foreword to the Student

Foreword to the Instructor

Foreword to the Student

**Chapter 1. Vectors and Matrices**

1. Vectors

2. Dot Product

3. Hyperplanes in

**Rn**

4. Systems of Linear Equations and Gaussian Elimination

5. The Theory of Linear Systems

6. Some Applications

**Chapter 2. Matrix Algebra**

1. Matrix Operations

2. Linear Transformations: An Introduction

3. Inverse Matrices

4. Elementary Matrices: Rows get Equal Time

5. The Transpose

**Chapter 3. Vector Spaces**

1. Subspaces of

**Rn**

2. The Four Fundamental Subspaces

3. Linear Independence and Basis

4. Dimension and Its Consequences

5. A Graphic Example

6. Abstract Vector Spaces

3. Linear Independence and Basis

4. Dimension and Its Consequences

5. A Graphic Example

6. Abstract Vector Spaces

**Chapter 4. Projections and Linear Transformations**

1. Inconsistent Systems and Projection

2. Orthogonal Bases

3. The Matrix of a Linear Transformation and the Change-of-Basis Formula

4. Linear Transformations on Abstract Vector Spaces

**Chapter 5. Determinants**

1. Properties of Determinants

2. Cofactors and Cramer’s Rule

3. Signed Area in

**R2**and Signed Volume in

**R2**

**Chapter 6. Eigenvalues and Eigenvectors**

1. The Characteristic Polynomial

2. Diagonalizability

3. Applications

4. The Spectral Theorem

**Chapter 7. Further Topics**

1. Complex Eigenvalues and Jordan Canonical Form

2. Computer Graphics and Geometry

3. Matrix Exponentials and Differential Equations

For Further Reading

Answers to Selected Exercises

List of Blue Boxes

Index

Answers to Selected Exercises

List of Blue Boxes

Index