{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe combine the orthogonal decomposition with Grassmann's formula.\n\n1. Let $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$. Let $W$ be a subspace of $V$, and let $W^\\perp$ be its [[orthogonal complement|079D68S3AW]].\n2. The subspace $W$ is [[finite-dimensional|0QW084460E]], since $V$ is. So [[$V = W \\oplus W^\\perp$|0SP7P0G7KQ]].\n3. By the definition of the [[direct sum|041JTZCK9Q]], $W + W^\\perp = V$ and the only way to write $0 = u + u'$ with $u \\in W$, $u' \\in W^\\perp$ is $u = u' = 0$.\n4. Let $x \\in W \\cap W^\\perp$. Then $x + (-x) = 0$ with $x \\in W$ and $-x \\in W^\\perp$. By step 3, $x = 0$. So $W \\cap W^\\perp = \\set{0}$.\n5. The set $W^\\perp$ contains $0$ and is closed under linear combinations, by linearity of the inner product in the first argument. So $W^\\perp$ is a subspace, and it is finite-dimensional by the same result as in step 2.\n6. By [[Grassmann's formula|06HTJNCMPA]], $\\dim(W + W^\\perp) + \\dim(W \\cap W^\\perp) = \\dim W + \\dim W^\\perp$, where $\\dim$ is the [[dimension|05G2G0P224]].\n7. By steps 3 and 4, the left side is $\\dim V + 0$. Hence $\\dim W + \\dim W^\\perp = \\dim V$.\n\n## Idea\n\nThe space splits as $W \\oplus W^\\perp$, and the two parts meet only in $0$. Dimensions add over such a splitting, as Grassmann's formula shows.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "15A03",
      "46C05"
    ],
    "name": "Proof of the formula for the dimension of the orthogonal complement",
    "proves": "0E2NYPVSP9",
    "sources": [
      {
        "locator": "Theorem 6.51",
        "url": "https://linear.axler.net/LADR4e.pdf"
      }
    ],
    "tag": "JSHFXYCSSK",
    "uses": [
      "079D68S3AW",
      "0QW084460E",
      "0SP7P0G7KQ",
      "041JTZCK9Q",
      "06HTJNCMPA",
      "05G2G0P224"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/JSHFXYCSSK/",
  "links": [
    {
      "field": "proves",
      "target": "0E2NYPVSP9"
    },
    {
      "field": "uses",
      "target": "079D68S3AW"
    },
    {
      "field": "uses",
      "target": "0QW084460E"
    },
    {
      "field": "uses",
      "target": "0SP7P0G7KQ"
    },
    {
      "field": "uses",
      "target": "041JTZCK9Q"
    },
    {
      "field": "uses",
      "target": "06HTJNCMPA"
    },
    {
      "field": "uses",
      "target": "05G2G0P224"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JSHFXYCSSK.md",
  "msc": [
    "15A03",
    "46C05"
  ],
  "name": "Proof of the formula for the dimension of the orthogonal complement",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JSH/JSHFXYCSSK.md",
  "status": null,
  "tag": "JSHFXYCSSK",
  "uri": "m2://JSHFXYCSSK",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-formula-for-the-dimension-of-the-orthogonal-complement/",
  "writing": {
    "tag": "JSHFXYCSSK",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}