# Low-rank error and explained variance

> ML preparation item.

- Stable ID: `original-la-low-rank`
- Area: Mathematics
- Topic: Linear Algebra
- Origin: original
- Expected time: 8 minutes
- Skills: svd, low-rank-approximation, pca-connection, matrix-norms
- Item JSON: https://mlprep.iwase.dev/data/items/original-la-low-rank.json

## Problem

A centered data matrix has singular values $5$, $2$, and $1$. Find the Frobenius error of its best rank-$1$ approximation and the fraction of total squared variation retained by the first principal component.


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