# Least squares, QR, and orthonormal projection

> ML preparation item.

- Stable ID: `mit-18-06-s2010-exam2-q2`
- Area: Mathematics
- Topic: Linear Algebra
- Origin: external
- Expected time: 18 minutes
- Skills: least-squares, normal-equations, qr, qr-factorization, projection, orthogonality
- Item JSON: https://mlprep.iwase.dev/data/items/mit-18-06-s2010-exam2-q2.json

This site does not reproduce the third-party problem.

- Source: MIT 18.06 Linear Algebra
- Document: Quiz 2 (listed by OCW as Exam 2)
- Locator: Question 2(a–c), page 2
- Official answer PDF page(s): 4
- Original problem: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/mit18_06s10_exam2_s10/
- Official answer: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/mit18_06s10_exam2_s10_soln/
- Answer status: official-solution
- Source caveat: The official solution's wording about dependent columns can blur the distinction between the unique fitted vector and potentially non-unique coefficient vectors.


## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `mit-18-06-s2010-exam2-q2`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.