{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0Y95Z7MT02"
    }
  ],
  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $T \\colon V \\to V$ be a linear map.\nLet $v_1, \\dots, v_k$ be [[eigenvectors|0N6960W0BV]] of $T$ for eigenvalues $\\lambda_1, \\dots, \\lambda_k$ that are pairwise distinct.\nThen $v_1, \\dots, v_k$ are [[linearly independent|0CRMGPAH83]].\n\n## Remarks\n\nThe proof is by induction on $k$: apply $T - \\lambda_k \\id_V$ to a linear relation, and the term of $v_k$ disappears.\nSo the sum of the eigenspaces of $T$ for distinct eigenvalues is a direct sum.\nIn particular, an endomorphism of an $n$-dimensional space has at most $n$ distinct eigenvalues.\n\n## Explanations\n\n### L1 human\n\nEigenvectors for different eigenvalues point in genuinely different directions.\nNone of them is a combination of the others.\nThis is the key fact behind diagonalisation.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "15A18"
    ],
    "name": "Eigenvectors with distinct eigenvalues are linearly independent",
    "role": "theorem",
    "sources": [
      {
        "locator": "Eigenspaces, geometric multiplicity, and the eigenbasis",
        "url": "https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors"
      }
    ],
    "tag": "0HNMFG8F8K",
    "uses": [
      "0B5H4208Z9",
      "0EVHJFTG9Z"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HNMFG8F8K/",
  "links": [
    {
      "field": "uses",
      "target": "0B5H4208Z9"
    },
    {
      "field": "uses",
      "target": "0EVHJFTG9Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HNMFG8F8K.md",
  "msc": [
    "15A18"
  ],
  "name": "Eigenvectors with distinct eigenvalues are linearly independent",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HN/0HNMFG8F8K.md",
  "status": null,
  "tag": "0HNMFG8F8K",
  "uri": "m2://0HNMFG8F8K",
  "url": "https://math2.tamnd.com/wiki/eigenvectors-with-distinct-eigenvalues-are-linearly-independent/",
  "writing": {
    "tag": "0HNMFG8F8K",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}