A Comprehensive Study of Vector Spaces and Subspaces in Linear Algebra

Vector spaces and subspaces are fundamental concepts of linear algebra that form the backbone of modern mathematics and its applications. This research paper presents a detailed and systematic study of vector spaces and subspaces, covering definitions, axioms, properties, examples, and major theorems. The paper also explores related concepts such as linear combinations, span, basis, dimension, linear transformations, quotient spaces, and applications in science and technology. This study aims to provide a comprehensive academic resource suitable for undergraduate, postgraduate students, and researchers.