Linear Algebra

Across many different fields, we face problems that need to be modeled, analyzed, and solved. These problems might involve electrical currents in electrical engineering, stresses and forces in mechanical engineering, sound waves in physics, images in computer vision, and, most importantly for us, machine learning.

Now imagine having a single language, a single set of tools, that allows us to work with all of these problems in a unified way. That language is linear algebra. Linear algebra gives us a framework for translating complex real-world systems into mathematical objects we can manipulate, analyze, and ultimately solve using well-defined calculations.

The power of linear algebra comes from how general it is. The same concepts apply across many domains, which is why it shows up everywhere. The downside of this generality is that the definitions can feel abstract or unintuitive at first. To address this, we’ll always connect abstract definitions to concrete examples, such as lists of numbers, tables of data, and geometric interpretations using arrows in space, so that strong intuition develops alongside formal understanding.

Formally speaking, linear algebra is the study of linear transformations on finite-dimensional vector spaces. While that may sound intimidating right now, there’s no need to worry. We’ll build everything from the ground up, starting with simple ideas and gradually layering on more powerful concepts. Each new idea will act like an upgrade, expanding the kinds of problems we’re able to tackle.

So, let’s begin at the foundation.

What exactly is a vector space?