# Linear Algebra with Applications

## Second EditionJeffrey Holt

©2017
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*Linear Algebra with Applications*, Second Edition, blends computational and conceptual topics throughout to prepare you for the rigors of conceptual thinking in an abstract setting. The early treatment of conceptual topics in the context of Euclidean space gives you more time, and a familiar setting, in which to absorb them. This organization also makes it possible to treat eigenvalues and eigenvectors earlier than in most texts. Abstract vector spaces are introduced later, once you have developed a solid conceptual foundation. Concepts and topics are frequently accompanied by applications to give you context and motivation. For those who learn best by example, *Linear Algebra with Applications* provides a large number of representative examples, over and above those used to introduce topics. The text also includes practice problems and over 2500 exercises, exposing you to computational and conceptual topics over a range of difficulty levels.

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Learn More## Table of Contents

**1. Systems of Linear Equations**

1.1 Lines and Linear Equations

1.2 Linear Systems and Matrices

1.3 Applications of Linear Systems

1.4 Numerical Solutions

**2. Euclidean Space**

2.1 Vectors

2.2 Span

2.3 Linear Independence

**3. Matrices**

3.1 Linear Transformations

3.2 Matrix Algebra

3.3 Inverses

3.4 LU Factorization

3.5 Markov Chains

**4. Subspaces**

4.1 Introduction to Subspaces

4.2 Basis and Dimension

4.3 Row and Column Spaces

4.4 Change of Basis

**5. Determinants**

5.1 The Determinant Function

5.2 Properties of the Determinant

5.3 Applications of the Determinant

**6. Eigenvalues and Eigenvectors**

6.1 Eigenvalues and Eigenvectors

6.2 Diagonalization

6.3 Complex Eigenvalues and Eigenvectors

6.4 Systems of Differential Equations

6.5 Approximation Methods

**7. Vector Spaces**

7.1 Vector Spaces and Subspaces

7.2 Span and Linear Independence

7.3 Basis and Dimension

**8. Orthogonality**

8.1 Dot Products and Orthogonal Sets

8.2 Projection and the Gram-Schmidt Process

8.3 Diagonalizing Symmetric Matrices and QR Factorization

8.4 The Singular Value Decomposition

8.5 Least Squares Regression

**9. Linear Transformations**

9.1 Definition and Properties

9.2 Isomorphisms

9.3 The Matrix of a Linear Transformation

9.4 Similarity

**10. Inner Product Spaces**

10.1 Inner Products

10.2 The Gram-Schmidt Process Revisited

10.3 Applications of Inner Products

**11. Additional Topics and Applications**

11.1 Quadratic Forms

11.2 Positive Definite Matrices

11.3 Constrained Optimization

11.4 Complex Vector Spaces

11.5 Hermitian Matrices

Glossary

Answers to Selected Exercises

Index