Why Linear Algebra for Machine Learning?

At its core, machine learning is about finding patterns in data. But for a computer to process information, whether it's a customer review, a stock price, or a photograph of a dog, that information must first be translated into a language it understands: the language of numbers. Linear algebra is the grammar of that language. It provides a powerful and efficient set of tools for organizing and manipulating these numbers, making it the computational foundation for virtually every machine learning model in existence.

Representing Data as Numbers

Before an algorithm can learn, we need a systematic way to structure our data. Let's consider a practical example: predicting house prices. The information for a single house might include its size (1,500 sq. ft.), the number of bedrooms (3), and its age (20 years). We can represent this house as an ordered list of numbers, which in linear algebra is called a vector:

$$\mathrm{house_vector} = [1500, 3, 20]$$

This vector is more than just a list. It's a point in a three-dimensional space, where each dimension corresponds to a feature. A different house, say one with 2,100 sq. ft., 4 bedrooms, and an age of 5 years, is simply another point in that same space: $[2100, 4, 5]$.

When we collect data for thousands of houses, we can stack these vectors together to form a grid of numbers. This grid is a matrix. Each row in our matrix is a house (a sample), and each column is a feature (size, bedrooms, age). This matrix becomes the central object that our machine learning model will work with.

Computing with Data Efficiently

Once our data is organized into vectors and matrices, we need to perform calculations. We might want to adjust model parameters, calculate the similarity between two items, or transform the data to highlight important patterns.

Without linear algebra, you would have to write loops to iterate through every number in your dataset, one by one. This is slow, inefficient, and makes the code difficult to read. Linear algebra provides a way to express complex computations across an entire dataset in a single line of code. Operations like matrix multiplication allow us to process thousands of data points simultaneously. This is not just a matter of convenience; it is the reason modern machine learning is computationally feasible. Libraries like NumPy are highly optimized to perform these vector and matrix operations at incredible speeds.

The Blueprint for Machine Learning Models

Many machine learning algorithms are not just supported by linear algebra; they are defined by it.

• Linear Regression: The classic problem of fitting a line to a set of data points is expressed as solving the equation $Ax = b$. Here, $A$ is the matrix of our data, $b$ is the vector of outcomes we want to predict (e.g., house prices), and solving for the vector $x$ gives us the parameters of our model.
• Neural Networks: A neural network is a series of interconnected layers. The process of passing data from one layer to the next involves a matrix-vector multiplication, which acts as a linear transformation, followed by an "activation function". Training a neural network is all about finding the right numbers for these matrices.
• Principal Component Analysis (PCA): This popular technique for dimensionality reduction finds the most important "directions" in the data. These directions are the eigenvectors of the data's covariance matrix, a central concept we will explore in Chapter 5.

The diagram below illustrates how raw data is transformed into a numerical format that machine learning algorithms, powered by linear algebra, can use to make predictions.

Raw information is converted into feature vectors, which are then collected into a data matrix. Machine learning algorithms use linear algebra to operate on this matrix and produce a final output or prediction.

In summary, learning linear algebra is not an optional academic exercise. It is the practical toolkit you need to understand how to represent data, how algorithms work under the hood, and how to implement them efficiently. In the sections that follow, we will begin building this toolkit from the ground up, starting with the simplest objects: scalars, vectors, and matrices.