Having established vectors and matrices as tools for representing and manipulating data, we now examine the underlying structure where these operations occur. This chapter focuses on vector spaces, the formal mathematical setting for groups of vectors. We will investigate the properties of these spaces and how they help us understand data structure and relationships.

Specifically, you will learn to define vector spaces and identify subspaces within them. We will explore how vectors combine to form linear combinations and define the span of a set of vectors. A key focus will be on linear independence, a concept important for understanding feature redundancy and selecting informative variables. Building on this, we'll define the basis of a vector space and its dimension, quantifying the fundamental components and 'size' of the space. We'll also look at important subspaces associated with matrices, namely the column space and null space, and understand the significance of matrix rank.

These concepts provide the theoretical foundation for analyzing feature vectors in machine learning, understanding model complexity, and preparing data for algorithms like dimensionality reduction. We will see how these abstract algebraic ideas relate directly to practical data analysis tasks.