Linear Algebra: A Comprehensive Analysis of Vector Spaces, Eigenvalues, and Matrix Theory

Abstract

This paper focuses on three fundamental pillars of linear algebra: vector spaces, eigenvalues, and matrix theory, providing both theoretical and numerical perspectives on these topics. The theoretical framework examines vector spaces through their axiomatic structure, subspaces, bases, dimensions, and linear transformations. It further investigates eigenvalues and eigenvectors as representations of invariant directions, emphasizing diagonalization, spectral properties, and the geometric interpretation of linear operators. Matrix theory is analyzed through matrix operations, determinants, invertibility, similarity transformations, canonical forms, and important decompositions, including LU, QR, and singular value decomposition (SVD). Furthermore, graph spectral analysis and low-rank SVD approximations highlighted the broad applicability of linear algebra in data science, numerical computation, network analysis, and applied mathematics.