{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0FWJ0VW561"
    },
    {
      "field": "uses",
      "target": "0JWM6GJNQW"
    },
    {
      "field": "uses",
      "target": "0RBT5PB74K"
    },
    {
      "field": "uses",
      "target": "0VT6P0GNQR"
    }
  ],
  "body": "\n## Definition\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $I$ be a set.\nA family $(e_i)_{i \\in I}$ of vectors of $V$ is *orthonormal* when, for all $i, j \\in I$,\n$$\\langle e_i, e_j \\rangle = \\delta_{ij},$$\nwhere $\\delta_{ij} = 1$ if $i = j$ and $\\delta_{ij} = 0$ if $i \\ne j$.\n\n## Remarks\n\nSo each $e_i$ has norm $1$, and $e_i$ and $e_j$ are [[orthogonal|0Y2X7P2WB7]] for $i \\ne j$.\nThe standard basis of $\\R^n$ is orthonormal for the dot product.\nThe functions $t \\mapsto e^{2\\pi i k t}$, $k \\in \\Z$, are orthonormal for $\\langle f, g \\rangle = \\int_0^1 f \\bar g\\, dt$.\n\n## Explanations\n\n### L1 human\n\nAn orthonormal family is like a set of coordinate axes: unit vectors at right angles.\nComputations with such vectors are simple, because all inner products are $0$ or $1$.\nThey are the best bases to work with in an inner product space.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0RW9TNNT87",
    "kind": "definition",
    "msc": [
      "15A63",
      "46C05"
    ],
    "name": "Orthonormal family",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Orthonormality"
      }
    ],
    "tag": "09VYHEF2VT",
    "uses": [
      "0TS7VZJBWD"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/09VYHEF2VT/",
  "links": [
    {
      "field": "defines",
      "target": "0RW9TNNT87"
    },
    {
      "field": "uses",
      "target": "0TS7VZJBWD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09VYHEF2VT.md",
  "msc": [
    "15A63",
    "46C05"
  ],
  "name": "Orthonormal family",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09V/09VYHEF2VT.md",
  "status": null,
  "tag": "09VYHEF2VT",
  "uri": "m2://09VYHEF2VT",
  "url": "https://math2.tamnd.com/wiki/orthonormal-family-definition/",
  "writing": {
    "tag": "09VYHEF2VT",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}