Linear Algebraic Framework of Quantum State Representation

Abstract

Linear algebra provides the mathematical foundation of quantum mechanics by rep-resenting physical states as vectors in complex Hilbert spaces and observables as linear operators. The concepts of vector spaces, basis, orthogonality, inner products, and matrix transformations enable a rigorous description of quantum systems. This paper presents a concise overview of the linear algebraic principles underlying quantum state representa-tion. The study discusses Hilbert spaces, Dirac notation, state normalization, superposition, the Born probability rule, Hermitian operators, and quantum measurement. Furthermore, the role of tensor products and matrix representations in quantum computing is briefly highlighted. An illustrative example demonstrates the application of these mathematical concepts in representing and analysing a quantum state. The paper emphasizes that linear algebra not only provides the language of quantum mechanics but also serves as the compu-tational framework for modern quantum technologies such as quantum computing, quantum information, and quantum communication. Keywords: Linear Algebra, Quantum Mechanics, Hilbert Space, Quantum State, Hermitian Operator, Superposition, Quantum Computing.