An Introduction to Linear Algebra: The Math I Slept Through in College That Now Powers AI
What’s old is new again. The math behind vectors, matrices, and tensors now powers modern AI. The same ideas that once felt abstract for me in the classroom now show up in recommendation systems, language models, customer segmentation, and modern marketing analytics.
Introduction
I have a confession to make. When I took the linear algebra class as an undergraduate at North Georgia College (now University of North Georgia) in the late 1980s, I had no idea I was learning math that would one day help power modern AI. At the time, it felt like one of the most impractical courses in the curriculum. I could multiply matrices, compute determinants, and survive a pop quiz on vector spaces, but I left class genuinely unsure why any of it mattered.
Looking back, I think that is true of a lot of things in life. You do not always appreciate them when they show up the first time. Sometimes their value only becomes obvious much later. The professor was not wrong. The textbook was not wrong. I just did not yet have the context to understand what those ideas would eventually make possible. I was a college kid trying to get through a math class. Relevance was not obvious.
Decades later, linear algebra turns out to be one of the core mathematical foundations of modern artificial intelligence. I wrote about this in a 2024 column for The Citizen Newspaper, reflecting on 45 years of STEM experience. It was a subject that “did not seem very relevant at the time,” but now clearly sits underneath the neural networks and machine learning systems behind today’s AI tools. When I returned to graduate school at Georgia Tech and started digging more seriously into machine learning, that became hard to ignore. I even took the Imperial College London Mathematics for Machine Learning linear algebra course on Coursera as a refresher, partly to fill in the gaps and partly to finally understand what I had missed the first time around. Better late than never.
This article is my attempt to give linear algebra the introduction it deserved when I first encountered it. My goal is not to turn you into a mathematician or walk you through pages of proofs. I want to give you enough intuition to understand what is happening when an AI system produces a result, whether that is a recommendation, a customer segment, an attribution model, or a text prediction. That kind of intuition matters more than most people realize. It helps you ask better questions, catch bad assumptions, and know when an output deserves confidence and when it deserves a second look.
Linear Algebra in Action
Let me show you what this math actually does before we get into definitions.
You have probably noticed that Netflix seems to know what you want to watch next, or that an ad shows up in your feed at exactly the right moment. That is not a coincidence, and nobody at those companies is manually reviewing your preferences. Something is running underneath all of that, and a big part of it is linear algebra.
Here is the basic idea. Every user gets represented as a long list of numbers. Watch history, time of day, what you skipped, what you finished, how you rated things. Every piece of content gets its own list of numbers too. I will define these lists properly in the next section and call them what they are: vectors. For now, just think of them as a numeric fingerprint.
To figure out whether you will like something, the system measures how similar your fingerprint is to the content’s fingerprint. Do that across millions of users and millions of titles at the same time and you need to organize everything into a grid, which is called a matrix, and run the whole calculation at once. That is matrix multiplication. The entire pipeline, from raw behavior data to the recommendation you see on screen, is a sequence of those operations.
In many modern systems, much of this work happens through math on organized lists of numbers rather than hand-written rules alone. Figure 1 shows the basic idea: users and content can be represented as vectors, and those relationships can be scaled across a larger user-item matrix.
There are two main flavors of this. User-Based Collaborative Filtering finds people similar to you and recommends what they liked. Item-Based Collaborative Filtering works the other way, recommending things similar to what you have already engaged with. For more technical details on this, including sample code, check out my earlier article, “UBCF vs. IBCF: Comparing Marketing Recommendation System Algorithms in R”.
Linear algebra concepts, like the example above, show up across all of marketing AI. Attribution models, customer segmentation, language tools, ad targeting. All of them are running some version of what I just described. So let us build the vocabulary to understand it properly.
What Is Linear Algebra?
Before we get into the building blocks, it helps to understand what the name actually means. Most introductions skip right past it, which is part of why the subject feels so abstract from the start.
Take the word “algebra” first. It refers, broadly, to the study of relationships between quantities. The algebra you learned in middle school explored relationships between unknown numbers. Linear algebra does the same thing, but with collections of numbers instead of individual ones.
“Linear” refers to a specific mathematical property. A relationship is linear if it is proportional and predictable. Double the input, double the output. That consistency is what makes linear relationships so useful computationally. Linear relationships can be broken apart, analyzed in pieces, and reassembled. Non-linear relationships do not behave that cleanly, which is one reason they are harder to analyze and work with at scale.
Put the two words “linear” and “algebra” together and you get this: linear algebra is the mathematics of predictable, structured transformations applied to organized collections of data. Kalid Azad at Better Explained frames it in a way I really like: “a matrix is essentially a shorthand for a spreadsheet of operations.” You are not just storing numbers. You are encoding a set of instructions for transforming them. I wish someone had put it that way in 1987. It would have saved me a lot of confusion.
With that framing in place, let us look at the actual objects linear algebra works with.
The Building Blocks of Linear Algebra
Linear algebra has four fundamental objects. Think of them as the nouns of the language. Everything else, the operations, the transformations, the AI applications, is based on what you can do with these four things.
- Scalar: A scalar is a single number. A price, a click-through rate, a customer’s age. Most of the individual numbers you see in a spreadsheet or a dashboard are scalars. Alone, they describe one attribute of one thing. The word scalar comes from the idea of scaling: multiply a vector or matrix by a scalar and every entry stretches or shrinks by that amount, as we will see shortly. Simple, but everything else builds from here.
- Vector: A vector is an ordered list of scalars. Think of a row in your spreadsheet, except that its order and direction matter mathematically. A customer profile might look like this: [age, average order value, days since last purchase, sessions this month]. That list is a vector. It does not just store data. It places that customer at a specific coordinate in a multi-dimensional space. Customers who behave similarly end up near each other in that space. Customers who are very different end up far apart. That geometric interpretation is not just a metaphor. It is the actual basis of customer segmentation, similarity matching, and the word embeddings I covered earlier in my NLP article.
- Matrix: A matrix is a two-dimensional grid of numbers, rows and columns. Think of your entire customer dataset as a matrix: each row is one customer, each column is one attribute. Matrices are where the real power of linear algebra shows up, because a matrix can represent not just data but operations and transformations. Multiplying one matrix by another can reveal hidden structure in the data in ways that are difficult to see otherwise. The weights inside a neural network are matrices. The pixels of an image are a matrix. The attention scores inside a language model are a matrix. We will come back to all of those.
- Tensor: This was not part of the vocabulary when I first learned about matrices in college, yet it is arguably one of the most important ones for understanding modern AI. A tensor is the generalization of a matrix to more than two dimensions. A grayscale image is a matrix, height by width. A color image is a three-dimensional tensor, height by width by three color channels. A batch of images processed together is a four-dimensional tensor. Major deep learning frameworks like PyTorch and TensorFlow are built around tensor operations because everything flowing through those systems is a tensor being transformed at each layer.
Those are the four building blocks of linear algebra. Next we will look at what you can actually do with them.
Core Linear Algebra Operations
Now that we have the building blocks, let us talk about what you can actually do with them. These are the verbs of the language. I am going to keep this practical and use simple numbers throughout. The goal is intuition, not a problem set.
Scalar Multiplication
Let us start with the simplest one. As we mentioned earlier a scalar is called a scalar because it scales. Multiply a vector by a scalar and every entry stretches or shrinks by the same amount, proportionally, predictably.
Say you have a customer described by three attributes: ad clicks, email opens, and purchases.
Customer A = [3, 5, 2]
Multiply by the scalar 2 and you get:
2 × [3, 5, 2] = [6, 10, 4]
Every attribute doubled. That predictability is exactly what makes scalar multiplication useful. It is the simplest form of a linear operation, and everything else builds from it.
Vector Addition
Adding two vectors means adding their corresponding entries. Same position, same attribute, added together.
Let’s use the original Customer A as follows:
Customer A = [3, 5, 2]
Say Customer B looks like this:
Customer B = [1, 3, 4]
Add Customer A and Customer B:
[3, 5, 2] + [1, 3, 4] = [4, 8, 6]
That combined vector might represent a household profile, a merged segment, or an intermediate step toward calculating an average across two groups. Simple concept, but it shows up everywhere.
One of my favorite examples of this comes from natural language processing. Word embeddings represent words as vectors, and you can do arithmetic on them to get meaningful results. The classic example is:
King − Man + Woman ≈ Queen
Let’s define the following:
King = [0.5, 0.7]
Man = [0.5, 0.2]
Queen = [0.3, 0.9]
Woman = [0.4, 0.3]
We can approximate Queen as follows:
[0.5, 0.7] - [0.5, 0.2] + [0.4, 0.3] = [0.4, 0.8]
That’s pretty close to the actual value of Queen = [0.3, 0.9]
That is just vector subtraction and addition applied to semantic representations of words. The fact that plain arithmetic on lists of numbers can capture something as abstract as the relationship between king and queen is one of the more remarkable things in modern AI. Seeing it graphically in Figure 4 makes this even more intuitive.
![Figure 4: Word embeddings represent words as vectors in a high-dimensional space. Simple vector arithmetic can capture semantic relationships: subtracting “man” from “king” and adding “woman” produces a result vector [0.4, 0.8] that lands close to “queen” at [0.3, 0.9]. The coordinate values shown here are illustrative, but the arithmetic reflects how modern language AI systems represent semantic similarity and relationships in vector space.](https://miro.medium.com/v2/resize:fit:1400/format:webp/1*eRftU3FirJ7ek-sd5_7U7Q.png)
In real word embedding systems, the dimensions number in the hundreds or thousands rather than just two, but the arithmetic works the same way. The geometry of the vector space captures semantic meaning, and simple addition and subtraction navigate that space in meaningful directions.
Dot Product
The dot product takes two vectors, multiplies their corresponding entries, and adds everything up into a single number. That number tells you how much the two vectors agree in direction and magnitude. Think of it as a similarity score. A high value means strong alignment. A value near zero means little or no alignment in that representation. A closely related measure, cosine similarity, normalizes for magnitude so that only the direction matters. Two vectors pointing the same way will have a cosine similarity of 1 regardless of their lengths. You will encounter cosine similarity often in clustering and NLP applications where the scale of vectors varies across observations.
Say a customer’s service usage can be represented as a vector with two attributes: email marketing engagement and SEO engagement.
Customer = [0.8, 0.3]
Compare that to two service packages:
Package 1 = [0.9, 0.4] (heavy email and SEO focus)
Package 2 = [0.1, 0.9] (light email, heavy social focus)
Dot product with Package 1:
(0.8 × 0.9) + (0.3 × 0.4) = 0.72 + 0.12 = 0.84
Dot product with Package 2:
(0.8 × 0.1) + (0.3 × 0.9) = 0.08 + 0.27 = 0.35
Package 1 is a much stronger match. The customer’s usage pattern aligns closely with Package 1 and only weakly with Package 2. That single number is doing a lot of work. Every time a recommendation engine decides your profile matches a piece of content, a dot product is involved somewhere in that decision.
See Figure 5 (below) for a geometric illustration of both comparisons.
Matrix Multiplication
Matrix multiplication is what happens when you need to compute dot products for many vectors at once. Instead of comparing one customer to one package at a time, you stack everything into matrices and get every result in a single operation.
Say we have two customers:
Customer A = [1, 0, 1]
Customer B = [0, 1, 1]
And two service packages written as columns:
Package 1 = [1, 0, 1]
Package 2 = [0, 1, 1]
Multiplying the customer matrix by the package matrix gives us one dot product per customer-package pair:
Customer A × Package 1: (1×1) + (0×0) + (1×1) = 2
Customer A × Package 2: (1×0) + (0×1) + (1×1) = 1
Customer B × Package 1: (0×1) + (1×0) + (1×1) = 1
Customer B × Package 2: (0×0) + (1×1) + (1×1) = 2
Result:
[[2, 1], [1, 2]]
Customer A fits Package 1 better. Customer B fits Package 2 better. Four answers, one operation. That is the power of matrix multiplication. When a neural network passes data from one layer to the next, matrix multiplication is one of the core operations taking place, transforming an entire batch of inputs simultaneously rather than one at a time.
Transpose
The transpose of a matrix flips it across its diagonal, swapping rows and columns. It is one of those operations that sounds trivial until you realize how often it shows up.
Take our customer matrix. As written, rows are customers and columns are services:
[[1, 0, 1], ← Customer A [0, 1, 1]] ← Customer B
Transpose it and the perspective flips. Now rows are services and columns are customers:
[[1, 0], ← Email [0, 1], ← Social [1, 1]] ← SEO
Now each row tells you which customers used a given service instead of which services a given customer used. That shift in perspective matters more than it might seem. When you solve a linear regression, the formula for the coefficients involves the transpose of the data matrix. One reason is dimensional: transposing aligns the matrix dimensions so the multiplication can proceed.
But it also produces something meaningful. The product X’X is a covariance matrix, capturing how your input variables relate to each other. Most people who run regressions every day have never thought about why that step is there. Now you know!
Advanced Linear Algebra Operations
The next two ideas are more specialized, but they show up directly in tools marketers use every day, so they are worth understanding at least conceptually. You do not need to compute these by hand. You do need to know what they are doing when your software runs them.
Inverse and Solving Systems of Equations
Think back to basic algebra. If you had the equation 3x = 12, you solved for x by dividing both sides by 3, or equivalently by multiplying by the inverse of 3, which is 1/3. Matrix inversion follows the same basic idea, just with matrices instead of single numbers.
In matrix form, a system of equations can be written like this:
A × x = b
Here, A is a matrix of known inputs, b is a vector of known outputs, and x is the vector of unknown values you are trying to solve for. When A is square and invertible, you can solve for x like this:
x = A⁻¹ × b
Here is a simple marketing example. Say you ran two campaigns and tracked conversions:
Campaign 1: spent [2, 1] on email and social, drove 5 conversions Campaign 2: spent [1, 3] on email and social, drove 7 conversions
In matrix form:
A = [[2, 1], ← Campaign 1 spend [1, 3]] ← Campaign 2 spend
b = [5, 7] ← Conversions
Solving A × x = b gives you x, the vector of channel effectiveness coefficients that would exactly fit those two observations. The figure below shows how this looks using our campaign example.
That is the basic intuition. In real regression problems, the setup is usually larger and messier, and the software typically solves a related system involving the design matrix rather than directly inverting a simple square matrix. Still, the core idea is the same: organize the data into matrices, then solve for the coefficients that best fit the observed outcomes.
The catch is that this only works cleanly when the inputs are not too similar to each other. When two variables are highly correlated, say paid search and paid social always move together in your data, the design matrix becomes ill-conditioned. This is a problem measurable by its condition number. A high condition number means small changes in your data can produce wildly different coefficient estimates. That is the linear algebra intuition behind multicollinearity. It is not a statistical quirk. It is a linear algebra problem.
Eigenvalues and Eigenvectors
These are the concepts that trip most people up, and honestly they tripped me up too when I first encountered them. Here is the intuition.
When you apply a matrix transformation to a vector, both its direction and magnitude usually change. Eigenvectors are the special vectors whose direction stays the same under that transformation. Their length may change, but they keep pointing along the same axis. The eigenvalue tells you by how much they were stretched or compressed.
A simple analogy helps. Imagine spinning a globe. Almost every point on the surface moves, but the axis of rotation stays fixed. That axis captures the intuition behind an eigenvector: a direction that remains unchanged by the transformation. Figure 9 shows all three of these ideas together.
Now bring that back to marketing data. Suppose you have customer data with two correlated variables, monthly ad spend and monthly revenue. In a cloud of data like that, some axes capture more of the variation than others. Eigenvectors identify those axes, and the corresponding eigenvalues tell you how much of the overall variation each one accounts for.
This is the idea behind Principal Component Analysis, or PCA. It is a technique that takes a dataset with many variables and finds a smaller set of axes that capture most of the meaningful structure. When you reduce a customer dataset from fifty variables down to five for segmentation, PCA is often the tool doing that work. You are not throwing data away randomly. You are keeping the strongest structure and letting go of the noise. The eigenvectors point the way, and the eigenvalues tell you which ones are worth keeping.
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That is why ideas that sound abstract in linear algebra class end up being so practical in real analytics work.
A Few More Ideas Worth Knowing
There are a few other linear algebra ideas worth knowing, even if only by name. Distance measures how close or far apart two vectors are, which is central to clustering, embeddings, and recommendation systems. Projection is the idea of mapping data onto a new axis or into a lower-dimensional space, which shows up in PCA and dimensionality reduction. Decomposition means breaking a matrix into simpler pieces, which is part of how recommendation engines and other AI systems uncover hidden structure in large datasets. We will see all three of these ideas show up again in the sections ahead.
Revisiting the Netflix Example: The Linear Algebra Behind the Recommendation
Earlier in this article, I described how Netflix recommends content. At the time, I kept the explanation intentionally vague because we did not yet have the vocabulary to go deeper. Now we do.
Let us go back and look at what is actually happening.
Every user on the platform can be represented as a vector, an ordered list of numbers encoding behavior across many dimensions: genres watched, completion rates, time of day, device type, and dozens more signals. Every piece of content can be represented as a vector encoding its characteristics. These are not simple lists. They are high-dimensional numeric fingerprints living in the same mathematical space, which is what makes comparison possible.
To measure how well a user and a piece of content match, the system often computes a dot product between the two vectors. A high result means strong alignment. A result near zero means little alignment in that representation. That single number, computed across millions of user-content pairs, is how candidate recommendations get scored and ranked.
To do that scoring across all users and all content simultaneously, the system organizes everything into matrices and runs the computation in one operation. Matrix multiplication is one of the core operations that makes that possible. What feels like an instant, personalized suggestion is the output of matrix calculations running at enormous scale.
In many recommendation systems, another step happens before or alongside similarity scoring: decomposition of the user-item interaction matrix into lower-dimensional representations that capture the most meaningful patterns while filtering out noise. The most common technique for this is Singular Value Decomposition, or SVD, which breaks a matrix into three simpler matrices that together reveal the hidden structure in user-item interactions. This is where projection comes in as well. Rather than working in the full original space, the system projects users and content into a compressed representation where the most informative structure is preserved. The result is a more manageable set of vectors that still captures what matters most about each user and each piece of content.
So the next time Netflix recommends something you end up loving, you now know what happened mathematically.
- A vector represented you.
- A vector represented the content.
- A dot product measured the match.
- Matrix multiplication helped scale it across the entire catalog.
- Decomposition and projection helped make the whole thing computationally tractable.
That is linear algebra at work in a product used by hundreds of millions of people every day.
How Linear Algebra Powers AI and Machine Learning
The Netflix example is a good illustration, but it is just one application. Most modern AI systems rely on the same mathematical machinery. Let me walk through the most important ones.
Neural Networks
A neural network can be seen as a long sequence of matrix multiplications. It takes those same matrix operations further than a recommendation system, chaining them together across many layers with a nonlinear activation function applied at each step. Each layer takes an input vector, multiplies it by a weight matrix, applies a simple nonlinear function, and passes the result to the next layer. That process repeats dozens or hundreds of times depending on the depth of the network. The weights stored in those matrices are what the network learns during training. When people say a large language model has billions of parameters, they mean the total number of values stored across all of its weight matrices.
Training the network means adjusting those matrix values to reduce prediction error. The forward pass through the network is driven by matrix multiplication. The backward pass uses derivatives and matrix operations to calculate how the weights should be adjusted. Calculus and linear algebra are both doing essential work at every step.
Dr. Gilbert Strang of MIT, whose linear algebra textbook and lecture series have shaped how generations of students learn the subject, has made the point throughout his career that understanding matrix structure is the key to understanding how modern computation scales. That observation has only become more true as neural networks have grown larger.
Large Language Models and Embeddings
When you type a prompt into a large language model, your text is first broken into tokens and those tokens are represented as vectors. Each token becomes a point in a high-dimensional space where similar meanings end up geometrically close to each other. That representation begins with an embedding, and it is a direct application of the ideas we covered in the building blocks section.
The attention mechanism inside transformer models, the architecture behind most modern language AI, takes this further. It computes relationships between tokens using learned query, key, and value projections, along with similarity scores that determine which parts of the input deserve the most attention. When a language model correctly resolves that “it” in a sentence refers to a specific noun from three sentences earlier, that resolution happened through geometric operations in a matrix space. Figure 12 illustrates this process using a simple marketing example.
If you want to go deeper on word embeddings and how NLP systems process language, I covered both in detail in my “Marketer’s Guide to NLP” article, including how text gets tokenized, how embeddings are built, and how models like these power the sentiment analysis and semantic search tools marketers use every day.
Dimensionality Reduction
Real-world datasets are high-dimensional. A customer dataset might have hundreds of variables. A language model might work in a space with thousands of dimensions. Feeding everything into a model at full dimensionality creates noise, computational expense, and instability.
Dimensionality reduction techniques, like PCA which we mentioned earlier, are based on eigenvalue decomposition and closely related linear algebra techniques. They identify the directions in the data that carry the most variation and project everything onto those directions. The result is a lower-dimensional representation that preserves the most important structure while discarding the noise.
Figure 13 shows this process using a simple three-dimensional example. This is not just a computational convenience. Removing noisy dimensions often produces better models, because you are keeping the signal and discarding the noise.
Optimization and Gradient Descent
Optimization sits at the intersection of calculus and linear algebra. Every AI model learns by minimizing a loss function, a measure of how wrong its current predictions are. The process of minimizing that function, called gradient descent, involves computing derivatives with respect to every parameter in the model. Those derivatives are organized into gradient vectors and matrices called Jacobians. Updating the model means adjusting the weight matrices in the direction that reduces the loss.
This is also why understanding the geometry of vector spaces matters for practitioners. A model that is not converging during training, or that is producing unstable results, often has a linear algebra explanation such as:
- The loss surface is poorly conditioned.
- The gradient directions are conflicting.
- The weight matrices have become ill-conditioned.
Knowing the vocabulary helps you diagnose problems rather than just observe them.
Generative AI and Image Models
Generative AI systems, including image generators, also rely on linear algebra at every stage. Images are tensors. The transformations applied during generation rely heavily on tensor operations, linear transformations, and learned vector representations. The latent space where the model navigates between concepts is a vector space. When you prompt a system to generate an image that blends two styles, the model is performing something like vector interpolation in that latent space, moving between two points and sampling from the space in between.
The same ideas, vectors, matrices, dot products, projections, and tensor operations, show up whether the output is text, images, audio, or recommendations. The applications look very different on the surface. The mathematics underneath is remarkably consistent.
Marketing Applications of Linear Algebra
Everything we have covered so far has been building toward this. The operations, the building blocks, and the AI systems all show up in the marketing tools you are probably already using. Let me walk through some of the most important applications.
Customer Segmentation and Clustering
When you segment customers, you are grouping points in a vector space. Each customer is represented as a vector of attributes including purchase history, demographics, and behavioral signals, and segmentation algorithms group customers whose vectors are close together in that space. Distance is the key operation here. Euclidean distance measures how far apart two vectors are in straight-line terms. Cosine similarity measures the angle between them rather than the distance, which makes it more useful when the magnitude, the overall length or scale of a vector, varies across customers. A high-spending customer and a low-spending customer might have identical purchasing patterns but very different vector lengths. Cosine similarity captures the similarity in pattern regardless of scale.
K-means clustering, one of the most common segmentation approaches, works by finding cluster centroids, which are just average vectors, and assigning each customer to the nearest one. Every iteration of the algorithm is vector arithmetic: computing distances, updating centroids, reassigning points. Figure 15 shows this process using a simple two-dimensional example.
PCA is frequently applied beforehand to reduce the dimensionality of the customer data, removing noisy variables and keeping the axes that explain the most variation. The segments you end up with are the result of linear algebra applied to your customer matrix.
Recommendation and Personalization
Recommendation engines, product suggestions, dynamic email content, and personalized web experiences all rely on embeddings, similarity scoring, and matrix operations. A user profile can be represented as an embedding, and candidate products or messages can be represented the same way. The system then scores the match between them using dot products, distance metrics, or small neural networks. There are two main approaches: user-based methods that find similar users and recommend what they liked, and item-based methods that find similar items and recommend what frequently goes together. What looks like personalization on the front end is often linear algebra running continuously in the background.
Figure 16 illustrates how both approaches work using a simple marketing services example. I explored this topic in my article on UBCF and IBCF recommendation algorithms, including full R code and side-by-side comparisons of both approaches applied to marketing data. The cosine similarity, correlation-based similarity measures, and related scoring methods used in those models all rely on the same broader mathematical ideas of representation, comparison, and similarity.
Natural Language Processing and Text Analytics
Modern text analytics tools represent words, sentences, and documents as vectors, placing them in a high-dimensional space where similar meanings end up geometrically close to each other. Sentiment analysis, brand mention classification, customer feedback categorization, and semantic search all operate in these embedding spaces. Linear algebra is what makes the geometry work. I covered the mechanics of this in detail in my NLP article if you want to go deeper.
When a tool tells you that 68% of this month’s customer reviews are positive, it has converted each review into a vector and classified it based on its position in a high-dimensional sentiment space. The words in the review, represented as embedding vectors, get combined into a document-level vector. That vector lands in a region of the space associated with positive or negative sentiment, and the classification follows from its position. The quality of that classification depends heavily on the quality of the embeddings, which are themselves the product of matrix operations applied during model training.
Figure 17 shows what this looks like in practice using familiar marketing vocabulary. Semantic search works similarly. When a customer types a query into a search bar, the query gets converted to a vector. The search engine finds documents whose vectors are most similar, using dot products or related similarity measures to score each candidate. In modern semantic search systems, vector geometry often does much of the heavy lifting, sometimes alongside more traditional keyword matching. The result is a search experience that captures meaning rather than just matching keywords, and linear algebra is a big part of what makes that possible.
Ad Targeting and Lookalike Audiences
When a platform builds a lookalike audience from your seed list, it is finding users whose embedding vectors are similar to the embeddings of your seed customers. The similarity measure is a dot product or a related distance metric computed across a high-dimensional user representation space. That space encodes signals like demographic attributes, browsing behavior, purchase history, content engagement, and app usage, often hundreds of dimensions representing everything the platform knows about each user.
Figure 18 illustrates how this works geometrically.
The quality of the lookalike depends on how well that embedding space captures meaningful dimensions of user behavior for your specific product. A user who looks similar in the platform’s embedding space may still have very different purchase intent if the dimensions that matter most for your business are not well represented there. A platform’s confidence score tells you how similar the audience is in its embedding space. It does not tell you whether that similarity is actually meaningful for your business. Understanding the linear algebra helps you ask better questions when evaluating audience quality rather than simply accepting the platform’s score at face value.
Marketing Mix Modeling
I covered MMM in depth in my four-part series, including hands-on implementations of Meta’s Robyn and Google’s Meridian. The short version is that at the core of most MMM approaches is a regression equation. And as we covered earlier in this article, regression is a linear algebra problem. The data you have been collecting about channel spend and business outcomes is not just a spreadsheet. It is a matrix waiting to be solved.
Here is what that looks like in practice. Your historical performance data gets organized into a design matrix where rows are time periods and columns are channel spend variables. Each row is a vector of inputs for that week or month. You are essentially solving a matrix equation of the form A × x = b, where A is your design matrix, b is your observed outcomes like revenue or conversions, and x is the vector of channel effectiveness coefficients you are trying to find. Those coefficients are what feed into the attribution numbers you review in your MMM dashboard every month.
Figure 19 shows how this data structure looks in matrix form.
Multicollinearity is one of the most common headaches in MMM, and most practitioners treat it as a statistical nuisance. It is actually a linear algebra problem. When two channels are highly correlated, say paid search and paid social always move together in your data, the design matrix becomes ill-conditioned. A high condition number means small changes in your data can produce wildly different coefficient estimates. That is not a statistical quirk. It is the math breaking down under the weight of redundant information. Ridge regression and other regularization techniques stabilize the solution by reducing that sensitivity.
This is why Robyn and Meridian sometimes produce channel rankings that seem counterintuitive, or why a coefficient flips sign when you add a new variable to the model. It is not a bug in the tool. The math has a reason, and now you know what it is.
Incrementality and Attribution
I covered incrementality measurement in my article on lift and attribution, “Measuring Lift: A Marketer’s Guide to Incrementality.” The statistical methods underlying incrementality testing, including difference-in-differences and regression-based lift models, all have linear algebra at their core. The observed outcomes can be organized as vectors, treatment effects can be expressed through vector differences, and the regression models used to control for confounding variables are matrix equations.
Understanding this helps you think more clearly about what these models are actually measuring and where their assumptions might break down. A lift model that produces a suspiciously clean result is worth scrutinizing. The linear algebra tells you where to look.
Forecasting and Predictive Modeling
Linear algebra also sits underneath many of the forecasting and predictive models marketers use to estimate demand, conversions, churn, and campaign outcomes. Whether the model is a basic regression, a regularized model like Ridge or LASSO, or part of a larger machine learning pipeline, the data is still organized into matrices and the coefficients or learned parameters are still being estimated through structured numerical operations on organized data.
Time series models like Meta’s Prophet make this especially visible. Prophet works by decomposing a forecast into separate components: trend, seasonality, holiday effects, and weekly patterns. Each component is estimated independently through matrix operations applied to the historical data, then recombined to produce the final forecast. The software handles all of that automatically, but the linear algebra is running underneath every prediction it makes.
Figure 20 shows what that decomposition looks like using website traffic data from a real forecasting project.
I covered this in detail in my article on forecasting website traffic for a local newspaper using Meta’s Prophet and R. The mechanics are more accessible than most people expect. Seeing the decomposition broken out by component is one of the best ways to build intuition for how these models actually work. The software may hide the math, but understanding what it is doing helps you ask better questions and catch problems when the forecast goes wrong.
Conclusion: The Math That Was Always There
I started this article with a confession. When I sat (aka “slept”) through linear algebra at North Georgia College in the late 1980s, I had no idea what I was learning would one day be part of the foundation of modern AI. It powers the tools I use every day as a practitioner and has become one of the most important subjects I would revisit as a graduate student at Georgia Tech decades later. It felt abstract and disconnected from anything practical at the time. It turned out to be foundational.
That is the thing about mathematics. Its value is not always obvious when you first encounter it. Linear algebra was there the whole time. The rest of us finally caught up to it.
If I could go back and talk to myself sitting in that classroom at North Georgia, staring at matrices and wondering when any of this would ever be useful, I would say: be patient. The subject that seems most abstract today has a way of becoming the most important thing you studied. As I noted in my STEM reflection, knowing how matrices and vectors work turned out to be a genuine competitive advantage when evaluating the strengths and weaknesses of different AI approaches decades later. Linear algebra is not just a math class. It is the grammar of modern AI, and AI is rapidly becoming the engine of modern marketing.
Why Intuition Matters
We are living in a moment when AI tools are proliferating faster than most practitioners can evaluate them. Dashboards produce confident outputs. Platforms report impressive numbers. Models generate attribution curves, customer segments, sentiment scores, and recommendations, all with the appearance of precision. But invisible does not mean unknowable. And that is the point of this article.
You do not need to derive eigenvectors by hand to benefit from understanding linear algebra conceptually. What you need is enough intuition to ask the right questions. Why are these two channels showing up as equally important when one clearly drove more incremental revenue? Why is the lookalike audience underperforming? Why does the sentiment tool keep miscategorizing this type of feedback? The answers often have a mathematical explanation. Knowing the vocabulary gives you a way in. The marketers who thrive in an AI-driven environment will not necessarily be the ones who operate the most tools. They will be the ones who understand what those tools are actually doing.
Key Takeaways
- Vectors represent things. Customers, products, documents, and campaigns can all be represented as numeric vectors. AI systems measure similarity, cluster groups, and make predictions by performing geometric operations on those vectors.
- Matrices represent relationships. The connection between channels and outcomes in a media mix model, the relationship between users and products in a recommendation system, the weights in a neural network: all of these are stored and manipulated as matrices.
- Dimensionality is a design choice. When you choose which variables to include in a model, you are defining the shape of the vector space the model operates in. More is not always better. Techniques like PCA can help you find the most informative lower-dimensional representation.
- Word embeddings are why modern NLP works. The ability to represent language as vectors and do arithmetic on those vectors to capture semantic meaning is one of the genuine achievements of modern AI. It is also why tools like semantic search, sentiment analysis, and content generation perform as well as they do.
- Your MMM results rest on linear algebra. The channel contribution and attribution numbers you review are downstream from a matrix equation. Understanding that helps you think more critically about model assumptions and results.
- Intuition compounds. Every concept you understand clearly becomes a foundation for the next one. The effort to build genuine understanding, rather than just accepting computational outputs, pays dividends that accelerate over time.
The most important next step is simply awareness. Now that you know what vectors, matrices, dot products, and embeddings are, you will start to notice them everywhere. In the tools you already use. In the results you already review. In the model outputs you already act on.
You do not need to do the math yourself. You need to know enough to recognize what the math is doing and whether it is being applied sensibly. In future articles, I plan to dig deeper into specific applications, including how word embeddings are built, how attention mechanisms work in transformers, and how dimensionality reduction applies directly to customer analytics.
For now, I hope this article has given you a new appreciation for a topic that I did not give nearly enough credit the first time around.
Resources
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