## Table of Contents

1. Systems of Linear Equations

1.1 Systems of linear equations

1.2 Matrices and linear systems

1.3 Elementary row operations

1.4 Echelon forms of a matrix

1.5 Solution set of a system of linear equations

1.6 Gaussian elimination

1.7 Gauss-Jordan elimination

1.8 Applications: Matrices in chemistry

1.9 Application: Electric circuits

2. Matrix Algebra

2.1 Matrix addition and scalar multiplication

2.2 Matrix multiplication

2.3 Matrix equations and linear systems

2.4 Inverse of a matrix

2.5 Solving a system using an inverse matrix

2.6 Elementary matrices

2.7 Block matrices

2.8 LU decomposition

2.9 Application: Leontief models

2.10 Application: Markov chains

3. Introduction to Vectors

3.1 Introduction to vectors

3.2 Vector operations

3.3 Dot product

3.4 Cross product

3.5 Application: 3D coordinate geometry

3.6 Application: Vectors in physics

4. Euclidean Vector Spaces

4.1 Vector spaces and subspaces

4.2 Spanning sets

4.3 Linear independence and dependence

4.4 Basis and dimension

5. Determinants

5.1 Introduction to determinants

5.2 Cofactor expansions

5.3 Application: Area and volume

5.4 Properties of determinants

5.5 Invertibility and determinants

5.6 Cramer’s rule

5.7 Permutations and determinants

6. General Vector Spaces

6.1 General vector spaces

6.2 Subspaces

6.3 Coordinatization

6.4 Four fundamental subspaces

6.5 Rank and nullity

7. Linear Transformations

7.1 Linear transformations between Euclidean spaces

7.2 General linear transformations

7.3 Isomorphisms

7.4 Rank and nullity of a linear transformation

7.5 Composition of linear transformations

7.6 Fundamental Theorem of Matrix Representations

7.7 Application: Transformations in 2D coordinate geometry

8. Eigenvalues and Eigenvectors

8.1 Eigenvalues and eigenvectors

8.2 Eigenspaces

8.3 Similarity and diagonalization

8.4 Complex eigenvalues and eigenvectors

8.5 Application: Inertia tensors

8.6 Application: Systems of first order differential equations

9. Inner Product Spaces and Orthogonality

9.1 Inner product spaces

9.2 Norms and distances

9.3 Orthogonal bases

9.4 Orthogonal complements

9.5 Orthogonal matrices

9.6 Singular value decomposition

9.7 Pseudoinverses

9.8 Complex inner product spaces

9.9 Application: Least-squares approximation

9.10 Application: Principal component analysis

10. Appendix: Notation

10.1 Notation

## A new and interactive introduction to linear algebra and matrix theory

**Linear Algebra** focuses on computation as a tool to build comprehension of this subject’s key principles. Definition-based topics are introduced to connect new concepts as the book progresses. A focus approach is taken on the foundational ideas of linear algebra. This book includes:

- 350+ auto-graded participation activities intended as reading exercises, such as question sets and animations
- Exceptionally visual presentations through animations
- 250+ end-of-section exercises for practice or homework
- Homework-appropriate, auto-graded, and randomized challenge activities in every section
- Dozens of applications that connect key concepts in linear algebra to real-world examples in physics, chemistry, circuits, and more

#### This zyBook replaces a traditional textbook. The modular approach taken by the authors allows instructors to fully customize by rearranging sections or including sections from other zyBooks, such as Algebra, Discrete Math, and Calculus.

#### Example of proof from Linear Algebra:

## What is a zyBook?

**Linear Algebra **is a web-native, interactive zyBook that helps students visualize concepts to learn faster and more effectively than with a traditional textbook. (Check out our research.)

Since 2012, over 1,700 academic institutions have adopted digital zyBooks to transform their STEM education.

### zyBooks benefit both students and instructors:

- Instructor benefits
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- Save chapters as PDFs to reference the material at any time

## Authors

**Chris Chan**

*Director, Content Development / M.A. in Mathematics / San Francisco State University*

**Alan Bass**

*Senior Content Developer, Mathematics / M.S. in Mathematics / University of North Carolina Wilmington*

**Susan Lauer**

*Content Developer, Mathematics / Ph.D. in Mathematics / Auburn University*