Basics of Linear Algebra for Machine Learning
Discover the Mathematical Language of Data in Python
Linear algebra is a pillar of machine learning. You cannot develop a deep understanding and application of machine learning without it.
In this new laser-focused Ebook written in the friendly Machine Learning Mastery style that you’re used to, you will finally cut through the equations, Greek letters, and confusion, and discover the topics in linear algebra that you need to know.
Using clear explanations, standard Python libraries, and step-by-step tutorial lessons, you will discover what linear algebra is, the importance of linear algebra to machine learning, vector, and matrix operations, matrix factorization, principal component analysis, and much more.
About the Ebook:
- PDF format Ebook.
- 6 parts, covering the main topics.
- 19 step-by-step tutorials.
- 211 pages.
- 92 Python (.py) code files included.
Clear and Complete Examples.
Designed for Developers. Nothing Hidden.
Click to jump straight to the packages.
Why Linear Algebra?
Linear algebra is a sub-field of mathematics concerned with vectors, matrices, and operations on these data structures. It is absolutely key to machine learning.
As a machine learning practitioner, you must have an understanding of linear algebra.
It’s a huge field of study that has made an impact on other fields, such as statistics, as well as engineering and physics. Thankfully, we don’t need to know the breadth and depth of the field of linear algebra in order to improve our understanding and application of machine learning.
Frustrated with the Math?
Have you ever been frustrated reading the description of a machine learning technique?
You’re reading along, things are going well, and then you hit an equation, and you are stopped in your tracks with questions like:
- … what do the terms mean?
- … why are there no operators between terms?
- … what does this Greek letter mean?
Unless you have a basic knowledge of linear algebra, you will not be able to read and understand even the most basic equations.
Have you heard of TensorFlow, Google’s Python library for deep learning?
Did you know that “tensor” is a term taken directly from the field of linear algebra and it simply means an array with more than two-dimensions?
Why Is Linear Algebra Important to Machine Learning?
So, why is linear algebra used so much to describe machine learning algorithms?
Linear algebra is about vectors and matrices and in machine learning we are always working with vectors and matrices (arrays) of data.
Linear algebra is essentially the mathematics of data.
It provides useful shortcuts for describing data as well as operations on data that we need to perform in machine learning methods.
Linear algebra is not magic
And, linear algebra is not trying to be exclusive or opaque.
As a first step, think of linear algebra as a shortcut language or notation to make describing some operations compact.
There are common machine learning techniques that can only be understood via their linear algebra descriptions because they come from the field. Techniques such as:
- Singular-Value Decomposition, or SVD.
- Principal Component Analysis.
- Linear Least Squares for Linear Regression.
Without a basic understanding of matrices and matrix operations, an understanding of these techniques will elude you.
The 3 Mistakes Made By Beginners
Once you discover the importance of linear algebra to machine learning, there are three key mistakes that beginners make:
1. Beginners Study Linear Algebra Too Early
If you ask how to get started in machine learning, you will very likely be told to start with linear algebra.
We know that knowledge of linear algebra is critically important, but it does not have to be the place to start. Learning linear algebra first, then calculus, probability, statistics, and eventually machine learning theory is a long and slow bottom-up path.
A better fit for developers is to start with systematic procedures that get results, and work back to the deeper understanding of theory, using working results as a context. I call this the top-down, or results-first, approach to machine learning, and linear algebra is not the first step, but perhaps the second or third.
2. Beginners Study Too Much Linear Algebra
When practitioners do circle back to study linear algebra, they learn far more of the field than is required for or relevant to machine learning.
Linear algebra is a large field of study that has tendrils into engineering, physics, and quantum physics. There are also theorems and derivations for nearly everything, most of which will not help you get better skill from, or a deeper understanding of, your machine learning model.
Only a specific subset of linear algebra is required, though you can always go deeper once you have the basics.
3. Beginners Study Linear Algebra Wrong
Linear algebra textbooks will teach you linear algebra in the classical university bottom-up approach. This is too slow (and painful) for your needs as a machine learning practitioner.
Like learning machine learning itself, take the top-down approach. Rather than starting with theorems and abstract concepts, you can learn the basics of linear algebra in a concrete way with data structures and worked examples of operations on those data structures. It’s so much faster (and fun).
Once you know how operations work, you can circle back and learn how they were derived. If you’re interested.
A Better Way into Linear Algebra
I am frustrated at seeing practitioner after practitioner diving into linear algebra textbooks and online courses designed for undergraduate students and giving up. The bottom-up approach is hard, especially if you already have a full time job.
Linear algebra is not only important to machine learning, but it is also a lot of fun, or can be if it is approached in the right way.
I want to help you see the field the way I see it: as just another set of tools we can harness on our journey toward machine learning mastery.
5 Areas of Linear Algebra to Focus On
You don’t need to know all of linear algebra.
The five key areas of linear algebra that I recommend you focus on are:
1. Learn Linear Algebra Notation
You need to be able to read and write vector and matrix notation.
Algorithms are described in books, papers, and on websites using vector and matrix notation.
Linear algebra is the mathematics of data and the notation allows you to describe operations on data precisely with specific operators.
You need to be able to read and write this notation.
2. Learn Linear Algebra Arithmetic
In partnership with the notation of linear algebra are the arithmetic operations performed.
You need to know how to add, subtract, and multiply scalars, vectors, and matrices.
A challenge for newcomers to the field of linear algebra are operations such as matrix multiplication and tensor multiplication that are not implemented as the direct multiplication of the elements of these structures, and at first glance appear nonintuitive.
3. Learn Linear Algebra for Statistics
You must learn linear algebra in order to be able to learn statistics. Especially multivariate statistics.
Statistics is concerned with describing and understanding data. As the mathematics of data, linear algebra has left its fingerprint on many related fields of mathematics, including statistics.
In order to be able to read and interpret statistics, you must learn the notation and operations of linear algebra.
4. Learn Matrix Factorization
Building on notation and arithmetic is the idea of matrix factorization, also called matrix decomposition.
You need to know how to factorize a matrix and what it means.
Matrix factorization is a key tool in linear algebra and used widely as an element of many more complex operations in both linear algebra (such as the matrix inverse) and machine learning (least squares, PCA, SVD, and more).
5. Learn Linear Least Squares
You need to know how to use matrix factorization to solve linear least squares.
Linear algebra was originally developed to solve systems of linear equations. These are equations where there are more equations than there are unknown variables. As a result, they are challenging to solve arithmetically because there is no single solution as there is no line or plane that can fit the data without some error.
Problems of this type can be framed as the minimization of squared error, called least squares, and can be recast in the language of linear algebra, called linear least squares.
If I could give one more reason, it would be: because it is fun.
Learning linear algebra, at least the way I teach it with practical examples and executable code, is a lot of fun. Once you can see how the operations work on real data, it is hard to avoid developing a strong intuition for the methods.
Introducing My New Ebook:
“Basics of Linear Algebra for Machine Learning“
Welcome to the “Basics of Linear Algebra for Machine Learning”
I designed this book to teach machine learning practitioners, like you, step-by-step the basics of linear algebra with concrete and executable examples in Python.
Who Is This Book For?
…so is this book right for YOU?
This book is for developers that may know some applied machine learning.
Maybe you know how to work through a predictive modeling problem end-to-end, or at least most of the main steps, with popular tools.
The lessons in this book do assume a few things about you, such as:
- You may know your way around basic Python for programming.
- You may know some basic NumPy for array manipulation.
- You want to learn linear algebra to deepen your understanding and application of machine learning.
This guide was written in the top-down and results-first machine learning style that you’re used to from Machine Learning Mastery.
What if I Am New to Machine Learning?
This book does not assume you have a background in machine learning.
That being said, I do recommend that you learn how to work through a predictive modeling problem first. It will give you the context for linear algebra. Otherwise the topic will feel too abstract.
What if I Am Just a Developer?
Perfect. I wrote this book for you.
What if My Math is Really Poor?
Maybe you learned linear algebra a long time ago back in school?
Maybe you never covered linear algebra before.
Perfect. This book is for you. I assume you know some basic arithmetic, and even then I give you a refresher.
What if I Am Not a Python Programmer?
You can handle this book if you are a programmer in another language, even if you are not experienced in Python.
Everything is demonstrated with a small code example that you can run directly.
All code is provided for you to play with, modify, and learn from.
I even show you how to manipulate NumPy arrays from first principles, because that is how we do linear algebra in Python.
The book even has an appendix to show you how to set up Python on your workstation.
What if I Am Working Through a Linear Algebra Course at a University?
This book is not a substitute for an undergraduate course in linear algebra or a textbook for such a course, although it is a great complement to such materials.
About Your Outcomes
…so what will YOU know after reading this book?
After reading and working through this book, you will know:
- What linear algebra is and why it is relevant and important to machine learning.
- How to create, index, and generally manipulate data in NumPy arrays.
- What a vector is and how to perform vector arithmetic and calculate vector norms.
- What a matrix is and how to perform matrix arithmetic, including matrix multiplication.
- A suite of types of matrices, their properties, and advanced operations involving matrices.
- What a tensor is and how to perform basic tensor arithmetic.
- Matrix factorization methods, including the eigendecomposition and singular-value decomposition.
- How to calculate and interpret basic statistics using the tools of linear algebra.
- How to implement methods using the tools of linear algebra such as principal component analysis and linear least squares regression.
This new basic understanding of linear algebra will impact your practice of machine learning.
After reading this book, you will be able to:
- Read the linear algebra mathematics in machine learning papers.
- Implement the linear algebra descriptions of machine learning algorithms.
- Describe your machine learning models using the notation and operations of linear algebra.
What Exactly Is in This Book?
This book was designed to be a crash course in linear algebra for machine learning practitioners. Ideally, those with a background as a developer.
This book was designed around major data structures, operations, and techniques in linear algebra that are directly relevant to machine learning algorithms.
There are a lot of things you could learn about linear algebra, from theory to abstract concepts to APIs. My goal is to take you straight to developing an intuition for the elements you must understand with laser-focused tutorials.
I designed the tutorials to focus on how to get things done with linear algebra. They give you the tools to both rapidly understand and apply each technique or operation.
Each tutorial is designed to take you about one hour to read through and complete, excluding the extensions and further reading.
You can choose to work through the lessons one per day, one per week, or at your own pace. I think momentum is critically important, and this book is intended to be read and used, not to sit idle.
I would recommend picking a schedule and sticking to it.
The tutorials are divided into five parts:
- Part 1: Foundation. Discover a gentle introduction to the field of linear algebra and the relationship it has with the field of machine learning.
- Part 2: NumPy. Discover NumPy tutorials that show you how to create, index, slice, and reshape NumPy arrays, the main data structure used in machine learning and the basis for linear algebra examples in this book.
- Part 3: Matrices. Discover the key structures for holding and manipulating data in linear algebra in vectors, matrices, and tensors.
- Part 4: Factorization. Discover a suite of methods for decomposing a matrix into its constituent elements in order to make numerical operations more efficient and more stable.
- Part 5: Statistics. Discover statistics through the lens of linear algebra and its application to principal component analysis and linear regression.
Below is an overview of the 19 step-by-step tutorial lessons you will work through:
Each lesson was designed to be completed in about 30-to-60 minutes by the average developer.
- Lesson 01: Introduction to Linear Algebra
- Lesson 02: Linear Algebra and Machine Learning
- Lesson 03: Examples of Linear Algebra in Machine Learning
- Lesson 04: Introduction to NumPy Arrays
- Lesson 05: Index, Slice, and Reshape NumPy Arrays
- Lesson 06: NumPy Array Broadcasting
- Lesson 07: Vectors and Vector Arithmetic
- Lesson 08: Vector Norms
- Lesson 09: Matrices and Matrix Arithmetic
- Lesson 10: Types of Matrices
- Lesson 11: Matrix Operations
- Lesson 12: Sparse Matrices
- Lesson 13: Tensors and Tensor Arithmetic
- Lesson 14: Matrix Decompositions
- Lesson 15: Eigendecomposition
- Lesson 16: Singular Value Decomposition
- Lesson 17: Introduction to Multivariate Statistics
- Lesson 18: Principal Component Analysis
- Lesson 19: Linear Regression
- Appendix A: Getting Help
- Appendix B: How to Set up a Workstation for Python
- Appendix C: Linear Algebra Cheat Sheet
- Appendix D: Basic Math Notation
You can see that each part targets a specific learning outcome, and so does each tutorial within each part. This acts as a filter to ensure you are only focused on the things you need to know to get to a specific result and do not get bogged down in the math or near-infinite number of digressions.
The tutorials were not designed to teach you everything there is to know about each of the theories or techniques of linear algebra. They were designed to give you an understanding of how they work, how to use them, and how to interpret the results the fastest way I know how: to learn by doing.
Table of Contents
The screenshot below was taken from the PDF Ebook. It provides you a full overview of the table of contents from the book.
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Enter your email address and your sample chapter will be sent to your inbox.
BONUS: Linear Algebra Python Code Recipes
…you also get 92 fully working Python scripts
Sample Code Recipes
Each recipe presented in the book is standalone, meaning that you can copy and paste it into your project and use it immediately.
- You get one Python script (.py) for each example provided in the book.
This means that you can follow along and compare your answers to a known working implementation of each example in the provided Python files.
This helps a lot to speed up your progress when working through the details of a specific task, such as:
- Creating and manipulating NumPy arrays.
- Arithmetic with vectors and matrices.
- Advanced matrix operations.
- Matrix factorization methods
- Statistical methods with matrices.
The provided code was developed in a text editor and is intended to be run on the command line. No special IDE or notebooks are required.
All code examples were designed and tested with Python 3.6+.
All code examples will run on modest and modern computer hardware and were executed on a CPU.
Python Technical Details
This section provides some technical details about the code provided with the book.
- Python Version: You can use Python 3.6 or higher.
- SciPy: You will use NumPy and scikit-learn.
- Operating System: You can use Windows, Linux, or Mac OS X.
- Hardware: A standard modern workstation will do.
- Editor: You can use a text editor and run the example from the command line.
Don’t have a Python environment?
The appendix contains step-by-step tutorials showing you exactly how to set up a Python machine learning environment.
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About The Author
Hi, I'm Jason Brownlee.
I live in Australia with my wife and son and love to write and code.
I have a computer science background as well as a Masters and Ph.D. degree in Artificial Intelligence.
I’ve written books on algorithms, won and ranked in the top 10% in machine learning competitions, consulted for startups and spent a long time working on systems for forecasting tropical cyclones. (yes I have written tons of code that runs operationally)
I get a lot of satisfaction helping developers get started and get really good at machine learning.
I teach an unconventional top-down and results-first approach to machine learning where we start by working through tutorials and problems, then later wade into theory as we need it.
I'm here to help if you ever have any questions. I want you to be awesome at machine learning.
What Are Skills in Machine Learning Worth?
Your boss asks you:
Hey, can you build a predictive model for this?
Imagine you had the skills and confidence to say:
...and follow through.
I have been there. It feels great!
How much is that worth to you?
The industry is demanding skills in machine learning.
The market wants people that can deliver results, not write academic papers.
Business knows what these skills are worth and are paying sky-high starting salaries.
A Data Scientists Salary Begins at:
$100,000 to $150,000.
A Machine Learning Engineers Salary is Even Higher.
What Are Your Alternatives?
You made it this far.
You're ready to take action.
But, what are your alternatives? What options are there?
(1) A Theoretical Textbook for $100+
...it's boring, math-heavy and you'll probably never finish it.
(2) An On-site Boot Camp for $10,000+
...it's full of young kids, you must travel and it can take months.
(3) A Higher Degree for $100,000+
...it's expensive, takes years, and you'll be an academic.
For the Hands-On Skills You Get...
And the Speed of Results You See...
And the Low Price You Pay...
Machine Learning Mastery Ebooks are
And they work. That's why I offer the money-back guarantee.
You're A Professional
The field moves quickly,
...how long can you wait?
You think you have all the time in the world, but...
- New methods are devised and algorithms change.
- New books get released and prices increase.
- New graduates come along and jobs get filled.
Right Now is the Best Time to make your start.
Bottom-up is Slow and Frustrating,
...don't you want a faster way?
Can you really go on another day, week or month...
- Scraping ideas and code from incomplete posts.
- Skimming theory and insight from short videos.
- Parsing Greek letters from academic textbooks.
Targeted Training is your Shortest Path to a result.
Professionals Use Training To Stay On Top Of Their Field
Get The Training You Need!
You don't want to fall behind or miss the opportunity.
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Generally, I would recommend starting with the book or topic that most interests you.
Nevertheless, one suggested order for reading the books is as follows:
- Linear Algebra for Machine Learning
- Master Machine Learning Algorithms
- Machine Learning Algorithms From Scratch
- Machine Learning Mastery With Weka
- Machine Learning Mastery With Python
- Machine Learning Mastery With R
- Time Series Forecasting With Python
- XGBoost With Python
- Deep Learning With Python
- Long Short-Term Memory Networks with Python
- Deep Learning for Natural Language Processing
I hope that helps.
Sorry, I do not have a license to purchase my books or bundles for libraries.
The books are for individual use only.
Multi-seat licenses create a bit of a maintenance nightmare for me, sorry. It takes time away from reading, writing and helping my readers.
If you have a big order, such as for a class of students or a large team, please contact me and we will work something out.
My best advice is to start with a book on a topic that you can use immediately.
Baring that, pick a topic that interests you the most.
If you are unsure, perhaps try working through some of the free tutorials to see what area that you gravitate towards.
Generally, I recommend focusing on the process of working through a predictive modeling problem end-to-end:
I have three books that show you how to do this, with three top open source platforms:
- Master Machine Learning With Weka (no programming)
- Master Machine Learning With R (caret)
- Master Machine Learning With Python (pandas and scikit-learn)
These are great places to start.
You can always circle back and pick-up a book on algorithms later to learn more about how specific methods work in greater detail.
Thanks for your interest.
You can see the full catalog of my books and bundles here:
Thanks for asking.
I try not to plan my books too far into the future. I try to write about the topics that I am asked about the most or topics where I see the most misunderstanding.
If you would like me to write more about a topic, I would love to know.
Contact me directly and let me know the topic and even the types of tutorials you would love for me to write.
Contact me and let me know the email address (or email addresses) that you think you used to make purchases.
I can look up what purchases you have made and resend purchase receipts to you so that you can redownload your books and bundles.
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The book “Deep Learning With Python” could be a prerequisite to”Long Short-Term Memory Networks with Python“. It teaches you how to get started with Keras and how to develop your first MLP, CNN and LSTM.
The book “Long Short-Term Memory Networks with Python” goes deep on LSTMs and teaches you how to prepare data, how to develop a suite of different LSTM architectures, parameter tuning, updating models and more.
The book “Long Short-Term Memory Networks With Python” focuses on how to implement different types of LSTM models.
The book “Deep Learning for Natural Language Processing” focuses on how to use a variety of different networks (including LSTMs) for text prediction problems.
The LSTM book can support the NLP book, but it is not a prerequisite.
Linux, MacOS, and Windows.
There are no code examples in “Master Machine Learning Algorithms“, therefore no programming language is used.
Algorithms are described and their working is summarized using basic arithmetic. The algorithm behavior is also demonstrated in excel spreadsheets, that are available with the book.
It is a great book for learning how algorithms work, without getting side-tracked with theory or programming syntax.
If you are interested in learning about machine learning algorithms by coding them from scratch (using the Python programming language), I would recommend a different book:
All of the books, except one, have been tested and work with Python 3 (e.g. 3.5 or 3.6).
Most of the books have also been tested and work with Python 2.7.
There is one book that required Python 2.7, that is:
This book will be updated for Python 3 soon.
Where possible, I recommend using the latest version of Python 3.
I do test my tutorials and projects on the blog first. It’s like the early access to ideas, and many of them do not make it to my training.
Much of the material in the books appeared in some form on my blog first and is later refined, improved and repackaged into a chapter format. I find this helps greatly with quality and bug fixing.
The books provide a more convenient packaging of the material, including source code, datasets and PDF format. They also include updates for new APIs, new chapters, bug and typo fixing, and direct access to me for all the support and help I can provide.
I believe my books offer thousands of dollars of education for tens of dollars each.
They are months if not years of experience distilled into a few hundred pages of carefully crafted and well-tested tutorials.
I think they are a bargain for professional developers looking to rapidly build skills in applied machine learning or use machine learning on a project.
Also, what are skills in machine learning worth to you? to your next project? and you’re current or next employer?
Nevertheless, the price of my books may appear expensive if you are a student or if you are not used to the high salaries for developers in North America, Australia, UK and similar parts of the world. For that, I am sorry.
I do offer discounts to students, teachers and retirees.
Please contact me to find out more.
I offer a ton of free content on my blog, you can get started with my best free material here:
About my Books
My books are playbooks.
They are intended for developers who want to know how to use a specific library to actually solve problems and deliver value at work.
- My books guide you only through the elements you need to know in order to get results.
- My books are in PDF format and come with code and datasets, specifically designed for you to read and work-through on your computer.
- My books give you direct access to me via email (what other books offer that?)
- My books are a tiny business expense for a professional developer that can be charged to the company and is tax deductible in most regions.
Very few training materials on machine learning are focused on how to get results.
The vast majority are about repeating the same math and theory and ignore the one thing you really care about: how to use the methods on a project.
Comparison to Other Options
Let me provide some context for you on the pricing of the books:
There are free videos on youtube and tutorials on blogs.
- Great, I encourage you to use them, including my own free tutorials.
There are very cheap video courses that teach you one or two tricks with an API.
- My books teach you how to use a library to work through a project end-to-end and deliver value, not just a few tricks
A textbook on machine learning can cost $50 to $100.
- All of my books are cheaper than the average machine learning textbook, and I expect you may be more productive, sooner.
A bootcamp or other in-person training can cost $1000+ dollars and last for days to weeks.
- A bundle of all of my books is far cheaper than this, they allow you to work at your own pace, and the bundle covers more content than the average bootcamp.
Sorry, my books are not available on websites like Amazon.com.
I carefully decided to not put my books on Amazon for a number of reasons:
- Amazon takes 65% of the sale price of self-published books, which would put me out of business.
- Amazon offers very little control over the sales page and shopping cart experience.
- Amazon does not allow me to contact my customers via email and offer direct support and updates.
- Amazon does not allow me to deliver my book to customers as a PDF, the preferred format for my customers to read on the screen.
I hope that helps you understand my rationale.
I am sorry to hear that you’re having difficulty purchasing a book or bundle.
I use Stripe and PayPal services to support secure and encrypted payment processing on my website.
Some common problems when customers have a problem include:
- Perhaps you can double check that your details are correct, just in case of a typo?
- Perhaps you could try a different payment method, such as PayPal or Credit Card?
- Perhaps you’re able to talk to your bank, just in case they blocked the transaction?
I often see customers trying to purchase with a domestic credit card that does not allow international purchases. This is easy to overcome by talking to your bank.
If you’re still having difficulty, please contact me and I can help investigate further.
I give away a lot of content for free. Most of it in fact.
It is important to me to help students and practitioners that are not well off, hence the enormous amount of free content that I provide.
You can access the free content:
I have thought very hard about this and I sell machine learning Ebooks for a few important reasons:
- I use the revenue to support the site and all the non-paying customers.
- I use the revenue to support my family so that I can continue to create content.
- Practitioners that pay for tutorials are far more likely to work through them and learn something.
- I target my books towards working professionals that are more likely to afford the materials.
All updates to the book or books in your purchase are free.
Books are usually updated once every few months to fix bugs, typos and keep abreast of API changes.
Contact me anytime and check if there have been updates. Let me know what version of the book you have (version is listed on the copyright page).
Please contact me anytime with questions about machine learning or the books.
One question at a time please.
Also, each book has a final chapter on getting more help and further reading and points to resources that you can use to get more help.
Yes, the books can help you get a job, but indirectly.
Getting a job is up to you.
It is a matching problem between an organization looking for someone to fill a role and you with your skills and background.
That being said, there are companies that are more interested in the value that you can provide to the business than the degrees that you have. Often, these are smaller companies and start-ups.
You can focus on providing value with machine learning by learning and getting very good at working through predictive modeling problems end-to-end. You can show this skill by developing a machine learning portfolio of completed projects.
My books are specifically designed to help you toward these ends. They teach you exactly how to use open source tools and libraries to get results in a predictive modeling project.