{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]]. First, $T(0) = 0$. Indeed, $T(0) = T(0 + 0) = T(0) + T(0)$, and subtracting $T(0)$ gives $T(0) = 0$.\n2. So $0$ lies in the [[kernel|0S23E6YE83]] $\\ker T$, and $\\{0\\} \\subseteq \\ker T$ always holds.\n3. For $u, v \\in V$, linearity gives $T(u) - T(v) = T(u - v)$. So $T(u) = T(v)$ if and only if $u - v \\in \\ker T$.\n4. Suppose $T$ is injective, and let $v \\in \\ker T$. Then $T(v) = 0 = T(0)$ by step 1, so $v = 0$ by injectivity. Hence $\\ker T = \\{0\\}$.\n5. Suppose $\\ker T = \\{0\\}$, and let $u, v \\in V$ with $T(u) = T(v)$. By step 3, $u - v \\in \\ker T$, so $u - v = 0$ and $u = v$. Hence $T$ is injective.\n6. By steps 4 and 5, $T$ is injective if and only if $\\ker T = \\{0\\}$.\n\n## Idea\n\nTwo vectors have the same image exactly when their difference lies in the kernel. So the map can only merge vectors if the kernel holds a nonzero vector.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "15A04"
    ],
    "name": "Proof that a linear map is injective iff its kernel is trivial",
    "proves": "0BJBYCGMWE",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Kernel_(linear_algebra)"
      }
    ],
    "tag": "M4VHXJSCD5",
    "uses": [
      "0NRYHJPQG2",
      "0S23E6YE83"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/M4VHXJSCD5/",
  "links": [
    {
      "field": "proves",
      "target": "0BJBYCGMWE"
    },
    {
      "field": "uses",
      "target": "0NRYHJPQG2"
    },
    {
      "field": "uses",
      "target": "0S23E6YE83"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/M4VHXJSCD5.md",
  "msc": [
    "15A04"
  ],
  "name": "Proof that a linear map is injective iff its kernel is trivial",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M4V/M4VHXJSCD5.md",
  "status": null,
  "tag": "M4VHXJSCD5",
  "uri": "m2://M4VHXJSCD5",
  "url": "https://math2.tamnd.com/wiki/proof-that-a-linear-map-is-injective-iff-its-kernel-is-trivial/",
  "writing": {
    "tag": "M4VHXJSCD5",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}