Geometric Residual Projection in Linear Regression: Rank-Aware Operators and a Geometric Multicollinearity Index

Residuals play a central role in linear regression, but their geometric structure is often obscured by formulas built from matrix inverses and pseudoinverses. This paper develops a rank-aware geometric framework for residual projection that makes the underlying orthogonality explicit. When the design matrix has codimension-one, the unexplained part of the response lies on a single unit normal to the predictor space, so the residual projector collapses to a rank-one operator nn^⊤ and no matrix inversion is needed. For general, possibly rank-deficient, designs the residual lies in a higher-dimensional orthogonal complement spanned by an orthonormal basis N, and the residual projector factorizes as NN^⊤. Using generalized cross products, wedge products, and Gram determinants, we give a basis-independent characterization of this residual space. On top of this, we introduce the Geometric Multicollinearity Index (GMI), a scale-invariant diagnostic derived from the polar sine that measures how the volume of the predictor space shrinks as multicollinearity increases. Synthetic examples and an illustrative real-data experiment show that the proposed projectors reproduce ordinary least squares residuals, that GMI responds predictably to controlled collinearity, and that the geometric viewpoint clarifies the different roles of regression projection and principal component analysis in both full-rank and rank-deficient settings.