I had to read “Linear Algebra and Its Applications” by David Lay for the Linear Algebra 1 class in my first semester in University. So this is a gentle introduction to Linear Algebra. The book doesn’t assume a lot of previous knowledge.

Author | David C. Lay |

ISBN-10 | 0201845563 |

**Chapter Structure**

Each chapter starts with an introductory example. Each section within a chapter ends with practice problems and exercises. Worked out examples with solutions are given too. As you would expect from a Linear Algebra book, there are lots of theorems and numerical notes.

**1. Systems of Linear Equations**

The first chapter gives some examples of linear systems. The row reduction algorithm is explained. I remember having to solve these kind of problems by hand for weeks. As is usual in mathematics, we learn to work out something with paper and pencil the hard way and then we figure out how to do it faster by writing a computer program. If you are into Python, please check out NumPy.

**2. Vector and Matrix Equations**

Chapter 2 starts with a number of examples as well. We learn about the fundamental idea of representing a linear combination of vectors as a product of a matrix and a vector. This leads to this famous equation:

A x = b

**3. Matrix Algebra**

Chapter 3 teaches about matrix operations such as matrix multiplication, matrix inversion and transposing matrices. The chapter ends with the Leontief Input Output Model from economics and applications to computer graphics.

**4. Determinants**

The introductory example in this chapter is about determinants in analytic geometry. Properties of determinants are mentioned as well as calculation methods.

**5. Vector Spaces**

I don’t know if it has anything to do with the chapter title, but the first example of this chapter is about space flight and control systems. In my opinion this chapter is more theoretical than the preceding chapters. The chapter ends with applications to difference equations and Markov Chains.

**6. Eigenvalues and Eigenvectors**

Dynamical systems and spotted owls are the topic of the introductory example of chapter 6. This chapter covers amongst others the characteristic equation, diagonalization and iterative algorithms to estimate eigenvalues.

**7. Orthogonality and Least Squares**

Chapter 7 begins with a short text about the North American Datum. After that we continue with sections on:

- orthogonality
- orthogonal sets
- orthogonal projections
- the Gram-Schmidt process
- least square problems
- inner product spaces

**8. Symmetric Matrices and Quadratic Forms**

A story about multi channel image processing is the introduction of chapter 8. This chapter has sections on quadratic forms and singular value decomposition.

The book is very readable and entertaining. The diverse list of examples are already reason enough to recommend “Linear Algebra and Its Applications”. I give this book 5 stars out of 5.