Vector space
A vector space is a set equipped with vector addition and scalar multiplication over a field that satisfy the vector-space axioms.
AI-Powered Search
A vector space is a mathematical space whose elements are vectors defined over a common set of dimensions.
- How it is used
- Provides shared coordinates for comparing search queries with documents in lexical and embedding-based retrieval.
- How deep it goes
- Defines dimensions, works through a 48-term query space, and explains shared embedding spaces and similarity.
- Examples
- Uses house attributes and an “apple juice” query mapped alongside two documents into a 48-word space.
- What you can do after
- Evaluate/understand it by checking that query and document embeddings share a space before comparing them for retrieval.
Natural Language Processing in Action, Second Edition
A vector space is a mathematical set of vectors that can be combined through operations such as addition and scalar multiplication.
- How it is used
- Frames vectors as computable representations for text similarity, distance calculations, and language-model operations.
- How deep it goes
- Explains coordinates, dimensionality, ordering, Euclidean geometry, dot products, and embedding analogies without developing formal axioms.
- Examples
- Uses latitude–longitude coordinates, graph-paper geometry, plotted term-frequency vectors, document vectors, and a sports-team analogy.
- What you can do after
- Apply vector representations to calculate similarity, preserve feature ordering, visualize low-dimensional points, and interpret embedding arithmetic.
Read how Natural Language Processing in Action, Second Edition explains vector space →
Generative AI in Action
A vector space is a mathematical set of vectors that supports operations such as addition and scalar multiplication.
- How it is used
- Supports semantic comparison and machine-readable representations for generative AI, search, classification, and language applications.
- How deep it goes
- Defines vectors, distances, and contextual meaning with examples, but defers detailed similarity algorithms and mathematical derivation.
- Examples
- Uses cat and dog vectors, embedding distances, an OpenAI Python call, and contextual representations of text and code.
- What you can do after
- Apply it via an embeddings API to convert text into vectors and inspect their dimensions and values.
Building Reliable AI Systems
A mathematical space whose elements are vectors and in which vector operations are defined.
- How it is used
- Uses a shared embedding space to retrieve relevant documents for retrieval-augmented generation.
- How deep it goes
- Connects shared embeddings to similarity search and Pinecone queries without deriving vector-space mathematics.
- Examples
- Uses a health-care policy chatbot to illustrate retrieval of physical-therapy coverage information.
- What you can do after
- Apply vector similarity through Pinecone by embedding queries with the document model and retrieving matching records.
Read how Building Reliable AI Systems explains vector space →
Related concepts
- embeddingproduced by · A numerical vector representation of an item or input that encodes features or relationships useful for computational tasks.
- similarity searchused by · Similarity search is a retrieval technique that finds or ranks items by comparing their representations with a query according to a similarity or distance measure.
- weighted averageuses · A weighted average is an average in which each value contributes according to an assigned numerical weight.
- weighted sumuses · A weighted sum is an expression formed by multiplying terms by corresponding scalar coefficients and adding the resulting products.
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