{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0SS770NS8B"
    }
  ],
  "body": "\n## Example\n\nLet $F$ be a field and let $n \\ge 0$ be an integer.\nLet $F[x]_{\u003cn}$ be the set of polynomials in $x$ with coefficients in $F$ and of degree less than $n$, with the zero polynomial included.\nIt is a [[vector space|000AYPA9AJ]] over $F$, with the usual addition of polynomials and multiplication by constants.\nThe monomials $1, x, x^2, \\dots, x^{n-1}$ form a [[basis|0KCQ9Y3QKA]], so $\\dim F[x]_{\u003cn} = n$.\n\n## Remarks\n\nThe set of polynomials of degree exactly $n$ is not a vector space.\nIt does not contain $0$, and it is not closed under addition.\nThe whole polynomial ring $F[x]$ is an infinite-dimensional vector space with basis $1, x, x^2, \\dots$.\n[[Differentiation|0SS770NS8B]] is a linear map from $F[x]_{\u003cn}$ to $F[x]_{\u003cn-1}$ for $n \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nPolynomials of small degree can be added and scaled, and the result stays a polynomial of small degree.\nSo they form a vector space.\nA polynomial $a_0 + a_1 x + a_2 x^2$ is the same data as the list $(a_0, a_1, a_2)$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "instance_of": [
      "000AYPA9AJ"
    ],
    "kind": "example",
    "msc": [
      "15A03"
    ],
    "name": "Space of polynomials of bounded degree",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Polynomial_ring"
      }
    ],
    "tag": "0JGZCDT5DN"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/0JGZCDT5DN/",
  "links": [
    {
      "field": "instance_of",
      "target": "000AYPA9AJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JGZCDT5DN.md",
  "msc": [
    "15A03"
  ],
  "name": "Space of polynomials of bounded degree",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JG/0JGZCDT5DN.md",
  "status": null,
  "tag": "0JGZCDT5DN",
  "uri": "m2://0JGZCDT5DN",
  "url": "https://math2.tamnd.com/wiki/space-of-polynomials-of-bounded-degree/",
  "writing": {
    "tag": "0JGZCDT5DN",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}