# Detect dependence and use a basis

> ML Prep screening item.

- Stable ID: `math-la-03-basis-coordinates`
- Collection: screening
- Area: Mathematics
- Topic: Linear Algebra
- Origin: original
- Difficulty: easy
- Expected time: 3 minutes
- Skills: linear-independence, basis-dimension, change-of-basis
- Item JSON: https://mlprep.iwase.dev/item/math-la-03-basis-coordinates.json

## Problem

In $\mathbb R^3$, let $v_1=(1,0,1)$, $v_2=(0,1,1)$, and $v_3=(1,1,2)$.
Determine whether the three vectors are independent. Give a basis and dimension
for their span, then give the coordinates of $x=(2,3,5)$ in your basis.


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## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-la-03-basis-coordinates`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.