People talk about AI like it’s magic.
“Chat-GPT knows me better than my wife!” “TikTok just read my mind!” “Grok is totally okay with my weird prompts!”
But what’s really happening under the hood? As much as it seems like it, it isn’t sorcery: it’s linear algebra - the math that gives order to chaos.
Sure, linear algebra isn’t glamorous. It won’t trend on social media as much as “hawk tuah” or the affairs of a CEO at a Coldplay concert.
And yet, we depend on it.
It’s the bedrock of AI. Every prediction, recommendation, and every “aha” moment is sitting on top of it.
Let’s meet this quiet genius running the show behind the curtain.
📈 The Geometry of Data
If Linear Algebra could say one thing, it would probably be:
Linear Algebra: “Whoooa, I’m spaced out, yo!”
Not what you expected? Let me explain.
Linear Algebra is fundamentally about spaces — multi-dimensional landscapes where each point represents a data point with numeric features.
For example, imagine a picture of a cat. If the picture is 256 by 256 pixels, that’s 65,536 numbers. Each image becomes a point in a 65,536-dimensional space.
Two cat photos will sit close together in that space. But a cat photo and a photo of a mountain will sit much further apart.
This giant “space” is where AI sees patterns, similarities, and differences — not through vision but through geometry.
If you imagine your data existing inside one of these spaces:
Each vector is a single point (a data item or observation).
Each matrix is an operation that moves, rotates, or reshapes these points.
Each linear transformation is how we bend that space to find patterns.
In Machine Learning, every input — an image, a sentence, or even your ex’s behavior online — becomes a coordinate within a vector space.
The “learning” part comes with mapping relationships between those coordinates.
🔍 Vectors - Data With Direction
A vector can be represented by a list of numbers (e.g., [4,5] or [1,2,3], etc.).
But that’s certainly not its limitation.
In fact, what a vector truly represents is a container for meaning in a particular space.
Take the phrase “machine learning is fun”. When that phrase is processed by an AI model, that sentence becomes a vector where each dimension represents some hidden quality: tone, context, semantic relationship, etc.
So what do you think it means if two sentences have very similar meanings? It means their corresponding vectors sit very closely within their space.
This is how language models like Chat-GPT “feel” similarity — not through definitions but through distance.
Mathematically, how similar two ideas feel is a measurement of how aligned their vectors are.
Understanding this idea of direction and alignment is key!
AI isn’t “comparing words”; it’s comparing positions in a conceptual landscape built entirely out of numbers.
🧠 Matrices - The Machinery Of Transformation
If vectors are the individual data points, matrices are the machines that shape them.
A matrix is essentially a rectangular grid of numbers:
If you think of a vector as a list of numbers, a matrix is like stacking many vectors side by side.
From a data perspective…
The rows often represent individual data points (like one image, one person, one sentence).
The columns often represent features (color intensity, age, number of words, etc.).
But just like how vectors are more than just a list of numbers, matrices are also more than just a grid — it’s a set of precise instructions for how a vector should change.
When you multiply a vector by a matrix, you’re performing a linear transformation: the vector may rotate, stretch, shrink, flip, or move into a completely different orientation within its space.
So what does this mean?
Well, this means one matrix might stretch dimensions related to brightness in an image! Another could rotate features in a sentence to make certain words or tones more prominent.
In essence, a matrix acts as a lens: when applied, it changes what parts of your data are in focus and what fades into the background.
🧭 How AI “Thinks” In Linear Algebra Terms
Step 1 — Representation. Whatever you give AI, whether it’s a sentence or an image, it gets turned into numbers called vectors.
Step 2 — Transformation. Those vectors are reshaped by matrices to highlight important patterns and relationships.
Step 3 — Mapping. The reshaped vectors are moved into new “spaces”.
Step 4 — Projection. The final vectors are compared against certain “directions” or categories to produce an answer, like “yup, that’s a dog”.
🔄 Seeing Linear Algebra In Action
Application | Linear Algebra Magic | Result |
|---|---|---|
Image Recognition | Convolutional layers perform matrix multiplication that scans for edges, textures, and patterns. | Cat vs croissant, solved. 🐱🥐 |
Natural Language Processing | Word embeddings represent semantic relationships in vector space; dot products reveal similarities between meanings. | Context-aware responses. |
Recommendation Systems | User and item vectors are multiplied in latent space to compute similarities. | “You’ll love this show”. |
Dimensionality Reduction | Eigenvectors identify high-variance directions in data to simplify while preserving structure. | Smaller, faster, clearer models. |
📘 Want To Understand This In Depth (Without Losing Your Mind?)
If this clicked — and you want to see how these ideas work step by step — my book Math For Machine Learning: Linear Algebra Simplified is your blueprint.
It doesn’t just cover vectors, matrices, and transformations.
It takes you through the entire toolkit — from spaces and dimensions to matrix factorization, eigenvalues, orthogonality, projections, and more — showing how each concept fuels real-world ML models.
You’ll see:
Core Linear Algebra concepts explained simply but in depth.
Exactly how they’re used in machine learning algorithms.
All in plain English, with examples that connect theory to practice.
Get it here → Math For Machine Learning: Linear Algebra Simplified
