# Linear transformations and quadratic expectations

> ML preparation item.

- Stable ID: `pml-book1-ex3.4`
- Area: Mathematics
- Topic: Statistics
- Origin: external
- Expected time: 15 minutes
- Skills: covariance, linear-transformation, trace, quadratic-form
- Item JSON: https://mlprep.iwase.dev/data/items/pml-book1-ex3.4.json

This site does not reproduce the third-party problem.

- Source: Probabilistic Machine Learning: An Introduction — online draft dated 2025-04-18
- Document: PML Book 1 online draft (2025-04-18), Chapter 3 exercises
- Locator: Exercise 3.4, book p. 104 (PDF p. 134)
- Official answer PDF page(s): 8
- Original problem: https://github.com/probml/pml-book/releases/download/2025-04-18/book1.pdf#page=134
- Official answer: https://probml.github.io/pml-book/solns-public.pdf#page=8
- Answer status: official-solution


## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `pml-book1-ex3.4`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.