Geometric Interpretation of Regression

Linear Algebra for Machine Learning · Axiom Academy

Visualizing projection onto the column space Linear regression can be understood as projecting the data vector onto the column space of : The projection of onto is given by: is idempotent: (projecting twice gives same result) The orthogonality of residuals to the column space is the core principle: Optimality: means we cannot improve the fit by adjusting any parameter Uniqueness: For full-rank , this condition has a unique solution Interpretation: The fitted model captures all patterns in the column space; residuals contain only unexplainable noise Generalization: Same principle applies to nonlinear regression, neural networks, and more advanced methods

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