# PCA direction and explained variance

> ML preparation item.

- Stable ID: `original-ml-pca`
- Area: Machine Learning
- Topic: Dimensionality Reduction
- Origin: original
- Expected time: 7 minutes
- Skills: pca-objective, centering, covariance-eigendecomposition, svd-pca, explained-variance
- Item JSON: https://mlprep.iwase.dev/data/items/original-ml-pca.json

## Problem

A centered two-dimensional dataset has covariance $\operatorname{diag}(9,1)$. Give the first principal direction, the fraction of variance it explains, and the reconstruction-error variance after projection to one dimension.


## 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-pca`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.