Extraction of linearly independent rows and dependency coefficients in datasets

This paper presents a constructive method for extracting a maximal set of linearly independent rows from a dataset matrix and computing explicit dependency coefficients for dependent rows. The proposed approach incrementally builds an independent subset by preserving linear independence at each step and subsequently derives the coefficient matrix expressing dependent rows as linear combinations of independent rows using a closed-form analytical expression based on the Gram matrix. Unlike classical rank-revealing techniques such as Gaussian elimination or singular value decomposition, which primarily identify independent components, the method directly provides explicit reconstruction relationships between dependent and independent rows. This enables structural analysis of redundancy within datasets and facilitates dimensionality reduction and feature selection. The mathematical formulation, algorithmic procedure, and numerical considerations are presented, demonstrating the effectiveness and interpretability of the approach.