{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "pml-book1-ex3.4",
    "area": "mathematics",
    "topic": "statistics",
    "origin": "external",
    "title": "Linear transformations and quadratic expectations",
    "skills": [
      "covariance-correlation"
    ],
    "priority": "supplementary",
    "difficulty": "medium",
    "estimated_minutes": 15,
    "prerequisites": [
      "expectation",
      "covariance",
      "matrix-multiplication"
    ],
    "source_id": "pml-book1-2022",
    "locator": {
      "document": "PML Book 1 online draft (2025-04-18), Chapter 3 exercises",
      "label": "Exercise 3.4",
      "page_label": "book p. 104 (PDF p. 134)",
      "solution_pages": "8"
    },
    "links": {
      "problem": "https://github.com/probml/pml-book/releases/download/2025-04-18/book1.pdf#page=134",
      "solution": "https://probml.github.io/pml-book/solns-public.pdf#page=8"
    },
    "selection_note": "Exercises the matrix identities behind transformed covariance and expected quadratic forms used throughout machine learning.",
    "answer_status": "official-solution"
  }
}