Mastering Vector Databases: Part II — The Core Concepts of Vector Embeddings and Similarity Search

2.1 Understanding Vector Spaces

2.1.1 Fundamentals of Vector Representations

At the heart of modern data processing is the concept of representing data as vectors. A vector is simply an ordered array of numbers that can describe a point in a high-dimensional space. This representation is especially useful for unstructured data — such as text, images, or audio — because machine learning models (like neural networks) can transform these inputs into dense vectors (or embeddings) that capture their essential characteristics.

Mathematically, if we have a vector v ∈ ℝⁿ, then each element v₁, v₂, …, vₙ represents a dimension in this space. The geometry of this space allows us to perform operations like addition, scaling, and dot products, which are fundamental to measuring similarity and performing clustering.

2.1.2 Linear Algebra Essentials

Key operations in vector spaces include:

Dot Product:
For two vectors u and v, the dot product is defined as:

• This operation not only measures similarity but also plays a role in projections and…