Section Article

Principles of Vector Spaces
Author(s): Dr. Aleena Grover

Abstract
This article examines the underlying concepts of vector spaces a core subject in linear algebra and mathematics in general. Vector spaces commonly referred to as linear spaces provide a structured framework for a range of mathematical and practical applications including solutions to systems of linear equations quantum physics and machine learning techniques. The article starts by providing a precise and formal description of vector spaces elucidating the essential features and axioms including closure associativity commutativity and the presence of an identity element and inverses. The text subsequently explores fundamental ideas like as subspaces linear independence bases and dimension. In addition the paper investigates the function of vector spaces in several disciplines emphasising their adaptability and significance in both theoretical and practical situations. The connection between vector spaces and linear transformations is examined with a focus on the practicality of using matrix representations. This study seeks to provide a thorough comprehension of vector spaces by conducting a meticulous analysis of these concepts. The ultimate goal is to facilitate their efficient utilisation in diverse fields.