Abstract

This paper explores the foundational principles of vector spaces a fundamental concept in linear algebra and mathematics at large. Vector spaces also known as linear spaces provide a framework for various mathematical and physical applications including solutions to systems of linear equations quantum mechanics and machine learning algorithms. The paper begins with a formal definition of vector spaces outlining the necessary properties and axioms such as closure associativity commutativity and the existence of an identity element and inverses. It then delves into key concepts including subspaces linear independence bases and dimension. Additionally the study examines the role of vector spaces in different fields highlighting their versatility and importance in both theoretical and applied contexts. The relationship between vector spaces and linear transformations is discussed emphasizing the utility of matrix representations. Through a detailed examination of these principles this pape