{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-la-basis-change",
    "area": "mathematics",
    "topic": "linear-algebra",
    "origin": "original",
    "title": "Basis, dimension, and coordinates",
    "skills": [
      "independence",
      "basis-dimension",
      "change-of-basis"
    ],
    "priority": "foundation",
    "difficulty": "easy",
    "estimated_minutes": 8,
    "prerequisites": [
      "linear-combinations"
    ],
    "prompt": "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.\n",
    "answer": "Since $v_3 = v_1 + v_2$, the set $\\{v_1,v_2\\}$ is a basis and the span has dimension $2$. Since $x = 2v_1 + 3v_2$, its coordinates in that basis are $(2,3)$.\n",
    "check": {
      "kind": "numeric",
      "id": "original-la-basis-change",
      "values": [
        2,
        2,
        3
      ]
    }
  }
}