{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "mit-18-06-s2010-exam3-q1",
    "area": "mathematics",
    "topic": "linear-algebra",
    "origin": "external",
    "title": "Symmetry, orthogonality, and eigenspaces",
    "skills": [
      "symmetric-matrices",
      "eigenvalues",
      "diagonalization"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 16,
    "prerequisites": [
      "outer-products",
      "orthogonal-complements"
    ],
    "source_id": "mit-18-06-s2010",
    "locator": {
      "document": "Quiz 3 (listed by OCW as Exam 3)",
      "label": "Question 1(a–d)",
      "page": 1,
      "solution_pages": "1"
    },
    "links": {
      "problem": "https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/mit18_06s10_exam3_s10/",
      "solution": "https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/mit18_06s10_exam3_s10_soln/"
    },
    "selection_note": "Uses a structured matrix to test proof, geometry, multiplicity, and trace in one compact question.",
    "answer_status": "official-solution"
  }
}