{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0JGB7A6GTS"
    },
    {
      "field": "uses",
      "target": "0MTBGVZ4XJ"
    }
  ],
  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *image* of $T$, also called the *range*, is\n$$\\im T = \\{T(v) : v \\in V\\} \\subseteq W.$$\n\n## Remarks\n\nThe image is a [[linear subspace|00YC1D652K]] of $W$.\nFor the map $x \\mapsto Ax$ of a matrix $A$, the image is the [[column space|03AJKXH1AV]] of $A$.\nThe map $T$ is surjective if and only if $\\im T = W$.\nThe dimension of the image is the [[rank|0ZZXSBTR4J]] of $T$.\n\n## Explanations\n\n### L1 human\n\nThe image of a linear map is everything that the map can produce.\nIf a map sends ordinary space onto a plane, then its image is that plane.\nThe image is always a subspace.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0WSFKKT6FF",
    "kind": "definition",
    "msc": [
      "15A04"
    ],
    "name": "Image of a linear map",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Image_(mathematics)"
      }
    ],
    "tag": "0JB228DCWB",
    "uses": [
      "0RJF668P1T"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0JB228DCWB/",
  "links": [
    {
      "field": "defines",
      "target": "0WSFKKT6FF"
    },
    {
      "field": "uses",
      "target": "0RJF668P1T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JB228DCWB.md",
  "msc": [
    "15A04"
  ],
  "name": "Image of a linear map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JB/0JB228DCWB.md",
  "status": null,
  "tag": "0JB228DCWB",
  "uri": "m2://0JB228DCWB",
  "url": "https://math2.tamnd.com/wiki/image-of-a-linear-map-definition/",
  "writing": {
    "tag": "0JB228DCWB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}