Linear algebra is quietly running every AI you use

Students ask me all the time what math they should take if they want to work in AI. They expect me to say calculus. Calculus matters, but the honest answer is a course most of them have never heard of: linear algebra. It's the math actually running ChatGPT, image generators, Netflix recommendations, and pretty much everything currently being called AI. And the core ideas are friendlier than the name suggests.

A matrix is a machine that transforms lists

Start with two humble objects. A vector is just a list of numbers. A matrix is a grid of numbers that acts like a machine: feed a list in, get a new list out. If you picture vectors as arrows on a graph, a matrix grabs the whole plane and stretches, squashes, rotates, or flips it in one move.

Here's the clever part. AI systems turn every word into a vector called an embedding, a list of hundreds or even thousands of numbers. Once words are points in space, meaning becomes geometry. Similar words sit near each other, and directions mean things. The classic example: take the vector for "king", subtract the vector for "man", add the vector for "woman", and the nearest word to where you land is "queen". Nobody programmed that. The direction for gender just shows up in the numbers, learned from reading a mountain of text.

ChatGPT is mostly multiplication

A large language model is layer upon layer of matrices transforming those word vectors, hundreds of times over, until the final list of numbers says which word probably comes next. Even "attention", the celebrated mechanism that lets these models track context, is built from dot products (the same dot product from Grade 12 Calculus and Vectors) plus yet more matrix multiplication.

Nothing inside is thinking in words. It's linear algebra running billions of times a second, and the intelligence, whatever it is, falls out of that.

Eigenvectors, the gentle version

When a matrix transforms the plane, most arrows get knocked onto a new direction. But a few special arrows keep their line: they come out stretched or squashed, sometimes flipped straight backwards, but never turned off the line they started on. Those directions are called eigenvectors, and the stretch (or flip) factor is the eigenvalue. That's the whole definition.

Why anyone cares: eigenvectors tell you what a matrix really does underneath all the shuffling. They're how engineers check whether a design will shake itself apart, how data scientists find the few directions that actually matter in a messy cloud of data, and how Google originally ranked web pages. If a matrix is a machine, the eigenvectors are its grain, the natural directions it works along.

The eigenslur (yes, really)

Which brings me to the strangest math word I've learned this year. Over the past several months a term called the "eigenslur" has been making the rounds online, spread mostly by a linguist on TikTok who goes by Etymology Nerd. It's more of a thought experiment than a published paper, but it sits on top of real research.

Here's the real part. People who study the insides of language models keep finding that abstract concepts live as directions in the model's space, the same way gender did in the king and queen example. Not just tidy concepts either. Researchers poking around in there have found directions for things like "this sentence is in French", "this is computer code", and, yes, "this word is being used as a slur". The eigenslur is the internet's tongue-in-cheek name for whatever would sit at the very far end of that last direction: a hypothetical word that is pure offense and nothing else. The "eigen" part is borrowed as a joke, because the idea distills something messy down to one clean direction, a bit like the way an eigenvector distills what a matrix does.

You don't need the spicy branding to see why it's interesting. If "being a slur" is a direction, then detecting it, filtering it, and steering a model away from it all start to look like linear algebra problems. A question about language and human harm turned into a question about vectors. That's where this math has landed in 2026.

Why this matters for an Ontario student

• It's coming sooner than you think. The vectors and dot products in Grade 12 Calculus and Vectors (MCV4U) are the trailer. Linear algebra proper is a first-year course, and it's required for engineering, computer science, math, and most data-flavoured programs.
• It's arguably the most employable math course you'll ever take. AI, computer graphics, data science, robotics, quantitative finance. All of them run on this one course.
• It clicks visually, not symbolically. The students who struggle usually only ever saw matrices as grids of numbers to grind through. The students who thrive have watched a matrix move the plane.

That last point is fixable in an afternoon. Our free grapher has matrix and vector modes, so you can type in a matrix and watch it act on the plane in real time. Spot the directions that refuse to turn and you've found the eigenvectors, no formulas required. And when linear algebra shows up on your first-year timetable, it's one of the courses I tutor (see subjects), because it's exactly where the jump from high school stings the most, and exactly the course that pays off the longest.