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# Linear Algebra

## A Geometric ApproachSecond Edition| ©2011 Ted Shifrin; Malcolm Adams

*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 t...

*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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*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.

Features

**Geometry is introduced early**, using vector algebra to do analytic geometry in the first section and dot product in the second.

**Concepts and understanding is emphasized, doing proofs in text and providing plenty of exercises.**To aid the student in adjusting to the mathematical rigor, blue boxes are provided which discuss matters of logic and proof technique or advice on formulating problem-solving strategies.

**Rotations, reflections, and projections are used as a first brush with the notion of linear transformation when introducing matrix multiplication. Linear transformations are then treated in concert with the discussion of projections.**Thus, the change-of-basis formula is motivated by starting with a coordinate system in which a geometrically defined linear transformation is clearly understood and asking for its standard matrix.

**Orthogonal complements are emphasized,**with their role in finding a homogenous system of linear equations that defines a given subspace of

*Rn*.

New to This Edition

**20% NEW exercises**have been added throughout the text to reinforce key concepts and give students practice in computation.

**Chapters have been updated, including:**

**New to Chapter 1, Vectors and Matrices:**new proof reasoning examples.

**New to Chapter 2, Matrix Algebra:**new sections on Linear Transformations and Elementary Matrices.

**New to Chapter 3, Vector Spaces:**streamlined treatment of four fundamental subspaces and clarified coverage of linear independence and basis.

**Updated and reorganized Chapters 4, Projections and Linear Transformations, and 5, Determinants**with improved clarity in the coverage of Change of Basis and the geometric material.

**Stronger emphasis throughout on key concepts and understanding,**through new proofs and a variety of text exercises.

**New Blue Boxes**, integrated throughout the text, discuss matters of logic and proof techniques or advice on formulating problem-solving strategies to aid the student in adjusting to the mathematical rigor.

**
Linear Algebra**

Second Edition| ©2011

Ted Shifrin; Malcolm Adams

# Digital Options

**Linear Algebra**

Second Edition| 2011

Ted Shifrin; Malcolm Adams

## 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

## Authors

### Ted Shifrin

**Theodore Shifrin**is a Professor of Mathematics and the Associate Head of the Mathematics Department at the University of Georgia. There, he has won multiple awards for teaching, including the Lothar Tresp Outstandin g Honors Professor Award in 2002 and 2010, as well as the Honoratus Medal in 1992. Professor Shifrin was one of five receipients of the University of Georgia's 1997 Josiah Meigs Award for Excellence in Teaching, and in 2000 he was given the Southeastern MAA Award for Distinguished College or University Teaching of Mathematics. In addition to Linear Algebra: A Geometric Approach, Professor Shifrin has published the textbooks

*Multivariable Mathematics: Linear Algebra*,

*Multivariable Calculus*, and

*Manifolds and Abstract Algebra: A Geometric Approach*, and he has also authored the

*Differential Geometry: A First Course in Curves and Surfaces*, a free, online text that is widely used all over the world. His research interests and publications have focused on integral geometry and complex algebraic geometry.

### Malcolm Adams

**Malcolm Adams**is a Professor of Mathematics and the Mathematics Department Head at the University of Georgia, where he also held the General Sandy Beaver Teaching Professorship from 2005-2008. He received is B.A. in Mathematics and Physics from the University of Oregon in 1978, and he earned his PhD in Mathematics from the Massachusetts Institute of Technology in 1982. Professor Adams's research interests focus on differential equations, especially in applications to biology and physics, and he has published another textbook,

*Measure Theory and Probability*, with Victor Guillemin. Outside of the university, he enjoys running, traveling, and hiking with his wife and three children.

**Linear Algebra**

Second Edition| 2011

Ted Shifrin; Malcolm Adams

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