linear algebra problem

Why Linear Algebra?

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I am in the process of learning data science. During a discussion with my father about what I was learning, he asked me a question that got me thinking. Why is linear algebra good for data science?

The answer is more complicated than it first appears. My first attempt to answer attempted to show a wide range of problems that linear algebra can solve. While this is part of the answer, the reasons we use linear algebra are considerably more profound.

One of the best books I have read thus far on the subject is by Jason Brownlee and is titled,  Basics of Linear Algebra for Machine Learning.

In the book’s preface, Brownlee points out, “Linear algebra is the mathematics of data. It’s all vectors and matrices of numbers. Modern statistics is described using the notation of linear algebra and modern statistical methods harness the tools of linear algebra.”

He goes on to note that modern machine learning techniques are described the same way, using the system of symbols and the tools drawn directly from linear algebra. He added that even some classical methods used in the domain including linear regression via linear least squares and singular-value decomposition are linear algebra procedures. Also, he notes that other processes, such as principal component analysis are a combination of linear algebra and statistics.

Why Linear Algebra for Data Science

The reasons for using linear algebra are at the heart of how computers work; doing addition subtraction, multiplication, and division. It even delves into how processors translate ones and zeros into characters, numbers, and data.

Arguably, this type of translation by itself is a kind of linear transformation that a computer accomplishes with linear algebra. In this way, and numerous others, computer programs are inherently good at solving linear algebra problems.

Linear algebra is particularly straightforward to turn into instructions such as the ones and zeros of machine code. It uses significantly fewer steps than it would take to translate calculus or differential equations.

Moreover, the math problems in these other branches of mathematics are frequently more easily converted into linear algebra before a computer can solve them.

On the other hand, calculus, while very elegant and able to solve a lot of mathematic problems is complicated for humans to learn. Calculus also requires considerably more steps to change into machine-readable instructions.

Differential equations, while requiring several more steps for computers to interpret into machine-readable code, can be somewhat easier to translate than calculus. However, it is not nearly as simple as translating linear algebra.

With computers, lists, and matrices are fundamental tools for solving linear algebra problems. Not coincidentally, they are also essential data types.

Linear Algebra Relatively Simple for Humans

Another reason for employing linear algebra also involves its relative simplicity for humans and programmers to understand. It serves as a somewhat straightforward interface between computers and computer coding languages such as python.

Perhaps more import than all of these reasons is that linear algebra can solve an enormous range of mathematical problems. These solvable problems extend from the simplest word problems in an eighth-grade math textbook to some of the most complex math in physics, chemistry, and biology.

And even when the problems are best solved with calculus, such equations are often translated first into linear algebra so that computers can come up with the answers. Also, importantly, this translation lets programmers, scientists, engineers, and statisticians more quickly devise computer programs that are used to solve such problems.

While many kinds of problems can be solved using linear algebra, some are inherently non-linear and may require other methods. I will touch on these at a later date.

Reference

Brownlee, J. Basics of Linear Algebra for Machine Learning. Edition v1.7. 2019.

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