1. In Module 2, we gave the normal equation (i.e., closed-form solution) for linear regression using MSE as the cost function. Prove that the closed-form solution for Ridge Regression is w = (1 + XT-X)-¹.XT.y, where I is the identity matrix, X = (x(¹), x(2),...,x(m)) is the input data matrix, x(¹) = (1,x₁,x₂,...,xn) is the i-th data sample, and y = (y(1), y(2),..., ym). Assume the hypothesis function h(x) = W₁ + W₁x₁ + W₂X₂ + ...+ WnXn, and y) is the measurement of hw(x) for the j-th training sample. The cost function of the Ridge Regression is E(w) = ₁(w²x) − y(¹) ² + 2₁w₁².

Fig: 1