Author(s): Johar M. Ashfaque
What is Linear Algebra
Linear Algebra is a branch of mathematics that concisely describes the coordinates and interactions of planes in higher dimensions and perform operations on them.
Think of it as an extension of algebra (dealing with unknowns) into an arbitrary number of dimensions. Linear Algebra is about working on linear systems of equations (linear regression is an example: y = Ax). Rather than working with scalars, we start working with vectors and matrices.
Broadly speaking, in linear algebra data is represented in the form of linear equations. These linear equations are in turn represented in the form of matrices and vectors.
— Vignesh Natarajan in answer to the question “How is Linear Algebra used in Machine Learning?”
As a field, it is useful to you because you can describe complex operations used in machine learning using the notation and formalisms from linear algebra.
Linear algebra finds widespread application because it generally parallelizes extremely well. Further to that most linear algebra operations can be implemented without messaging passing which makes them amenable to MapReduce implementations.
— Raphael Cendrillon in answer to the question “Why is Linear Algebra a prerequisite behind modern scientific/computational research?”
The Minimum Linear Algebra for Machine Learning
Linear Algebra is a foundation field that is to say that the notation and formalisms are used by other branches of mathematics to express concepts that are also relevant to machine learning.
For example, matrices and vectors are used in calculus, needed when you want to talk about function derivatives when optimizing a loss function. They are also used in probability when you want to talk about statistical inference.
…it’s used everywhere in mathematics, so you’ll find it used wherever math is used…
— David Joyce, in answer to the question, “What is the point of linear algebra?”
The minimum linear algebra that should be learned to improve your capabilities in machine learning would have to be the following 3 topics:
- Notation: Knowing the notation will let you read algorithm descriptions in papers, books, and websites to get an idea of what is going on. Even if you use for-loops rather than matrix operations, at least you will be able to piece things together.
- Operations: Working at the next level of abstraction in vectors and matrices can make things clearer. This can apply to descriptions, code, and even thinking. Learn how to do or apply simple operations like adding, multiplying, inverting, transposing, etc., matrices and vectors.
- Matrix Factorization: Especially matrix deposition methods like SVD and QR. The numerical precision of computers is limited, and working with decomposed matrices allows you to sidestep a lot of the overflow/underflow madness that can result. Also, a quick LU, SVD, or QR decomposing using a library will give you ordinary least squares for your regression problem. A bedrock of machine learning and statistics.
If you want to get into the theory of it all, you need to know linear algebra. If you want to read white papers and consider cutting edge new algorithms and systems, you need to know a lot of math.
— Jesse Reiss in answer to the question “How important is linear algebra in computer science?”
5 Reasons To Improve Your Linear Algebra
Of course, you can dive deeper.
If you need to know more and get better does not motivate you down the path, here are five reasons that might give you that push.
- Building Block: Linear algebra is absolutely key to understanding the calculus and statistics you need in machine learning.
- Deeper Intuition: If you can understand machine learning methods at the level of vectors and matrices, you will improve your intuition for how and when they work.
- Get More From Algorithms: A deeper understanding of the algorithm and its constraints will allow you to customize its application and better understand the impact of tuning parameters on the results.
- Implement Algorithms From Scratch: You require an understanding of linear algebra to implement machine learning algorithms from scratch. At the very least, to read the algorithm descriptions and, at best, to effectively use the libraries that provide the vector and matrix operations.
- Devise New Algorithms: The notation and tools of linear algebra can be used directly in environments like Octave and MATLAB, allowing you to prototype modifications to existing algorithms and entirely new approaches very quickly.
Linear Algebra will feature heavily in your machine learning journey, whether you like it or not.
2 Video Courses To Learn Linear Algebra
Here are some suggestions.
1. Linear Algebra Refresher
This is a quick whip around the topics in linear algebra you should be familiar with.
The video is titled “Linear Algebra for machine learning” and was created by Patrick van der Smagt using slides from University College London.
2. Linear Algebra Crash Course
The second suggestion is the Linear Algebra crash course presented as an optional module in Week 1 of his Coursera Machine Learning course.
This is suited to the engineer or programmer who is perhaps less or not at all familiar with linear algebra and is looking for a first bootstrap into the topic.
It contains 6 short videos, and you can access a YouTube playlist here titled “Machine Learning — 03. Linear Algebra Review”.
The topics covered include:
- Matrices and Vectors
- Addition and Scalar Multiplication
- Matrix-Vector Multiplication
- Matrix Matrix Multiplication
- Matrix Multiplication Properties
- Inverse and Transpose
Programming Linear Algebra
As a programmer or engineer, you likely learn best by doing.
As such, you may wish to grab a programming environment or library and start coding up matrix multiplication, SVD, and QR decompositions with test data.
Below are some options you might like to consider:
- Octave: Octave is the open-source version of MATLAB, and for most operations, they are equivalent. These platforms were built for linear algebra. This is what they do, and they do it very well. They are a joy to use.
- R: It can do t, but it's less beautiful than Octave. Check out this handy report: “Introduction to linear algebra with R” (PDF)
- SciPy numpy.linalg: Easy and fun if you are a Python programmer with clean syntax and access to all the operations you need.
- BLAS: Basic Linear Algebra Subprograms like multiplication, inverse, and the like. Ported or available in most programming languages.
- LAPACK: Linear Algebra Library, the successor to LINPACK. The place to go for various matrix factorizations and the like. Like BLAS, ported or available in most programming languages.
Linear Algebra Books
This section lists some of the top textbooks on Linear Algebra for beginners.
These are some beginner textbooks that cover the foundations of linear algebra:
These are books that lean more towards the application of linear algebra:
- Numerical Linear Algebra by Lloyd Trefethen.
- Linear Algebra and Its Applications by Gilbert Strang.
- Matrix Computations by Gene Golub and Charles Van Loan
Published via Towards AI