Having established the mechanics of vectors, matrices, and their operations, we now connect these mathematical tools to their direct use in machine learning. The previous chapters focused on the definitions and computations of linear algebra. This chapter demonstrates how those components are applied to prepare data and build models.

You will see how several foundational machine learning tasks are fundamentally problems of linear algebra. We will cover how:

• A dataset of features and observations is structured as a matrix for efficient processing.
• The entire problem of linear regression can be expressed compactly with the matrix equation $Ax = b$.
• Eigenvectors are used in Principal Component Analysis (PCA) to reduce the dimensionality of data.
• The dot product serves as a direct measure of similarity between vectors, a frequent operation in search and recommendation algorithms.

By the end of this chapter, the connection between abstract mathematical objects and concrete machine learning applications will be clear.