Matrix Applications in NLP: Vector Databases Simplified with FAISS and GRADIO

Vector Databases (VDBs) are now an essential building block in Natural Language Processing (NLP), facilitating the efficient storage and retrieval of high-dimensional semantic embeddings. Linear algebra is at the core of these systems, where matrix operations form the basis of embedding creation, similarity computation, and indexing. We discuss the mathematical underpinnings of VDBs from a matrix-based formulation in this paper. We show how similarity measures like cosine similarity and distance metrics like Euclidean and Mahalanobis distances drive nearest-neighbor retrieval. With FAISS for indexing and Gradio for prototyping, we introduce a comparative examination of exact vs. approximate search approaches in NLP retrieval tasks. Our work is a hybrid approach that unites mathematical precision with practical assessment, closing the gap between abstract matrix derivations and interactive use of VDBs.