10. Linear Regression via Vector Projection | Linear Algebra Made Easy скачать в хорошем качестве

10. Linear Regression via Vector Projection | Linear Algebra Made Easy

What does linear regression mean geometrically? Why is least squares an orthogonal projection? Viewing linear regression through geometry reveals that it is not merely about fitting a line, but about finding the point in a subspace that is closest to the data vector. In this video, linear regression is interpreted as an orthogonal projection. In ℝⁿ, the observed data vector y is projected onto the subspace spanned by the vectors 1 and x₁. The resulting projection is the predicted vector ŷ, while the residual ε = y − ŷ lies entirely in the orthogonal complement of that subspace, which explains why Xᵀε = 0. From this perspective: • ŷ is exactly the orthogonal projection of y • ε represents the perpendicular error component • The regression coefficients (b₀, b₁) are simply the coordinates of ŷ in the basis [1, x₁] If linear regression once felt like a collection of formulas, this geometric view reveals its true essence: linear regression is an orthogonal projection, and least squares is the algebra that computes its coordinates. -------------------------------------------------- 📘 Open-source books & Python code: https://github.com/Visualize-ML 🎬 Full playlist:    • Linear Algebra Made Easy   ☕ Support the project: https://buymeacoffee.com/drginger_jiang 📩 Contact: jiang.visualize.ml@gmail.com -------------------------------------------------- Voice: 生姜 DrGinger Video: 崔崔 CuiCui Library: Manim Thanks: Tsinghua University Press, 3Blue1Brown -------------------------------------------------- This video is part of Linear Algebra Made Easy (Iris Book Series), an open-source project dedicated to building geometric intuition for linear algebra, statistics, and machine learning. We create visual explanations to make mathematics simple, elegant, and accessible. If this video helped something click, consider liking 👍 sharing, and subscribing 🔔 You can also support the project via ❤️ Super Thanks or ☕ Buy Me a Coffee. English & Chinese subtitles available.