# Basis, dimension, and coordinates

> ML preparation item.

- Stable ID: `original-la-basis-change`
- Area: Mathematics
- Topic: Linear Algebra
- Origin: original
- Expected time: 8 minutes
- Skills: independence, basis-dimension, rank-nullity, change-of-basis
- Item JSON: https://mlprep.iwase.dev/data/items/original-la-basis-change.json

## Problem

In $\mathbb{R}^3$, let $v_1 = (1,0,1)$, $v_2 = (0,1,1)$, and $v_3 = (1,1,2)$. Determine a basis and dimension for their span. Express $x = (2,3,5)$ in your chosen basis.


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## Tutoring contract

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