{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.\nThen the [[product|0EX2HPXPJ6]] $V \\times W$ is finite-dimensional, and\n$$\\dim(V \\times W) = \\dim V + \\dim W,$$\nwhere $\\dim$ is the [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nLet $(v_i)$ be a basis of $V$ and let $(w_j)$ be a basis of $W$.\nThen the vectors $(v_i, 0)$ and $(0, w_j)$ together form a basis of $V \\times W$.\nFor example, $\\dim(\\R^2 \\times \\R^3) = 5$.\nBy induction, $\\dim(V_1 \\times \\cdots \\times V_k) = \\dim V_1 + \\cdots + \\dim V_k$.\n\n## Explanations\n\n### L1 human\n\nA vector of $V \\times W$ is a pair, so we need the directions of $V$ and the directions of $W$.\nThe dimensions add up.\nThe plane is the product of two lines, and $1 + 1 = 2$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "15A03"
    ],
    "name": "Dimension of a direct product",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Direct_sum_of_modules"
      }
    ],
    "tag": "06HN47VPTH",
    "uses": [
      "0MM085RX00",
      "05G2G0P224"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/06HN47VPTH/",
  "links": [
    {
      "field": "uses",
      "target": "0MM085RX00"
    },
    {
      "field": "uses",
      "target": "05G2G0P224"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/06HN47VPTH.md",
  "msc": [
    "15A03"
  ],
  "name": "Dimension of a direct product",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/06H/06HN47VPTH.md",
  "status": null,
  "tag": "06HN47VPTH",
  "uri": "m2://06HN47VPTH",
  "url": "https://math2.tamnd.com/wiki/dimension-of-a-direct-product/",
  "writing": {
    "tag": "06HN47VPTH",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}