10A. Linear Regression Is Just Projection! | Least Squares, Subspaces & Closed-Form Solution

In this lecture, we build the intuition behind Least Squares and Linear Regression using Linear Algebra and geometry. We begin with vector projection and then extend the idea to projecting a vector onto the Column Space of a matrix. From there, we understand how any target vector can be viewed as a combination of a component inside the Column Space and an error component perpendicular to it. That geometric idea naturally leads to the concept of Least Squares. We then connect the residual error to the Left Null Space, derive the logic behind the Normal Equation, and finally understand the closed-form solution of Linear Regression. The lecture follows this flow: Vector Projection → Projection onto Subspaces → Column Space → Orthogonality → Residual Error → Least Squares → Normal Equation → Closed-Form Linear Regression The goal is to understand why Linear Regression works, rather than simply memorizing a formula. Topics covered: Linear Algebra, Vector Projection, Subspaces, Column Space, Left Null Space, Orthogonality, Least Squares, Residual Error, Normal Equation, Closed-Form Solution, Linear Regression, Machine Learning Mathematics #LinearAlgebra #LinearRegression #LeastSquares #MachineLearning #MathForAI #DataScience