## Making the Complex Simple

Linear algebra is typically defined as the branch of mathematics dealing with vector spaces, linear equations, and linear transformations. These items are important because linear equations are very easy to solve (once you know how to do it!). Thus, they give efficient approximations for more complex equations. This holds true even for whole systems of equations in more dimensions than you can really picture in your mind, making it vital for some of the most important fields today such as quantum mechanics. And even something as everyday as compressing an image into a .jpeg file format; yup, that’s linear algebra. This Introduction to Linear Algebra book will help you master the basics in the field so you can continue on to great things.

## Introduction to Linear Algebra: What’s Inside

### Chapters 1-4: Foundations of Linear Algebra

In the first chapter of Introduction to Linear Algebra you’ll find a very friendly step-by-step introduction to linear equations. This includes linear systems of equations, matrices and vectors, and solutions to linear systems. Throughout the book you’ll find loads of examples, and the only material you’ll really need to know beforehand is Calculus 1. Although, more calculus knowledge wouldn’t hurt.

Chapter 2 then talks about linear transformations. It starts by talking about vector spaces, which you’ll need to know in order to grasp how transformations work. This chapter then talks about transformations in terms of linearity properties, and about applying them in terms of geometry. For instance, transformations can rotate or shift an object.

Chapter 3 gets into the basics of subspaces and their dimensions. The next chapter then expands into the more abstract ideas of arbitrary linear spaces. Having the basics chapter is great for undergraduate students just getting familiar with linear algebra. The more in-depth and abstract chapter is then a great resource for graduate students. It goes a step above subspaces, but without going into full blown graduate material. This makes it a great foundation for graduate students who can then build upon it with more advanced classes.

### Chapters 5-8: Adding Angles and Leaving Matrices Behind

The topic of orthogonality in Chapter 5 lets you look at angles rather than just lines. Most of the chapter has material covered in standard undergrad linear algebra courses and can serve as a good supplement for them. The last part of the chapter might not be covered in undergrad courses, letting you expand into higher-level areas.

Chapter 6 then covers determinants and trace, while Chapter 7 looks at Eigenvalues and Eigenvectors. Chapter 8 covers canonical forms where you start with a transformation rather than matrices. It relates to Chapter 7, but once again taking a more abstract perspective. So, whether you’re learning about these topics for the first time or wanting to dig deeper into the subject, you’ll find useful material in this Introduction to Linear Algebra.

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