# Ridge normal equations and effect

> ML preparation item.

- Stable ID: `original-ml-ridge`
- Area: Machine Learning
- Topic: Regression
- Origin: original
- Expected time: 8 minutes
- Skills: linear-regression, normal-equations, regularization, model-complexity
- Item JSON: https://mlprep.iwase.dev/data/items/original-ml-ridge.json

## Problem

Derive the minimizer of

$$\lVert X\mathbf{w}-\mathbf{y}\rVert_2^2+\lambda\lVert\mathbf{w}\rVert_2^2$$

for $\lambda>0$. State how increasing $\lambda$ typically changes coefficient magnitude and training fit.


## Answer policy

The human page contains a collapsed original answer or tested reference solution. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-ml-ridge`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.