You've seen how vectors represent individual data points or features, and how matrices can organize entire datasets or represent linear transformations. Now, we'll connect these concepts to a common task in machine learning: finding the optimal parameters for a model. Very often, this task boils down to solving a system of linear equations.

Consider one of the most fundamental models: linear regression. The goal is to predict a target value $y$ based on a set of input features $x_1, x_2, ..., x_n$ . The model assumes a linear relationship:

y \approx \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \dots + \theta_n x_n

Here, $\theta_0, \theta_1, \dots, \theta_n$ are the model parameters (or weights, coefficients) that we need to determine based on the training data. If we have $m$ data points, we can represent the features as a matrix $X$ (where each row is a data point, often with an added column of 1s for the intercept term $\theta_0$ ) and the target values as a vector $y$ . The parameters form a vector $\theta$ .

The objective in linear regression is typically to minimize the sum of squared differences between the predicted values and the actual values. Calculus and linear algebra show that the optimal parameter vector $\theta$ that achieves this minimization satisfies the normal equations:

X^T X \theta = X^T y

Look closely at this equation.

$X^T X$ results in a square matrix (let's call it $A$ ).
$\theta$ is the vector of unknown parameters we want to find (let's call it $x$ , matching the chapter's notation).
$X^T y$ results in a vector (let's call it $b$ ).

Suddenly, the problem of training a linear regression model transforms into solving the familiar matrix equation:

A x = b

This is exactly the form $Ax=b$ that this chapter focuses on. Finding the optimal parameters for linear regression requires solving this system for the unknown vector $x$ (which represents $\theta$ ).

This pattern isn't limited to simple linear regression.

Regularized Regression: Techniques like Ridge Regression modify the equation slightly to prevent overfitting, often leading to systems like $(X^T X + \lambda I) \theta = X^T y$ , where $\lambda$ is a regularization parameter and $I$ is the identity matrix. This is still fundamentally an $Ax=b$ problem, just with a different matrix $A$ .
Optimization Algorithms: Even for more complex models where solutions aren't found through a single matrix equation, solving linear systems often appears as a sub-problem within iterative optimization algorithms (like Newton's method) used during the training process.

Therefore, understanding how to represent and solve systems of linear equations like $Ax=b$ is not just a theoretical exercise in linear algebra. It's a practical requirement for implementing and comprehending the mechanics behind several important machine learning algorithms. Having established why solving these systems is relevant in machine learning, the following sections will explore the methods used to find the solution vector $x$ , starting with the concept of the matrix inverse.