{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe compute the entries of $M'$ directly.\n\n1. Write $\\mathcal{B} = (b_1, \\dots, b_n)$ and $\\mathcal{B}' = (b'_1, \\dots, b'_n)$, and let $P = (p_{ij})$.\n2. By the definition of the [[change of basis matrix|0KWXYZPJKN]], $b'_k = \\sum_{i=1}^n p_{ik} b_i$ for each $k$.\n3. By the definition of the [[matrix of a bilinear form|0P24YW565F]], $M_{ij} = B(b_i, b_j)$ and $M'_{kl} = B(b'_k, b'_l)$.\n4. By step 2, $M'_{kl} = B\\bigl(\\sum_i p_{ik} b_i, \\sum_j p_{jl} b_j\\bigr)$.\n5. A [[bilinear form|0P799D6AMN]] is linear in each argument. Expanding both arguments gives $M'_{kl} = \\sum_{i=1}^n \\sum_{j=1}^n p_{ik}\\, B(b_i, b_j)\\, p_{jl}$.\n6. By step 3 and the definition of the [[transpose|0W809K6AW0]], $p_{ik} = (P^T)_{ki}$. So $M'_{kl} = \\sum_{i,j} (P^T)_{ki} M_{ij} P_{jl}$.\n7. The right side is the $(k, l)$ entry of the product $P^T M P$, by the definition of matrix multiplication.\n8. This holds for all $k$ and $l$, so $M' = P^T M P$.\n\n## Idea\n\nEach new basis vector is a combination of old ones, with coefficients in a column of $P$. Bilinearity expands $B(b'_k, b'_l)$ into a double sum of old entries. That double sum is exactly a matrix product with $P$ on the right and $P^T$ on the left.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "15A63",
      "15A21"
    ],
    "name": "Proof of the change of basis rule for a bilinear form",
    "proves": "0SPYAGE7T0",
    "sources": [
      {
        "locator": "Theorem 2.8",
        "url": "https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf"
      }
    ],
    "tag": "RATQ474V7S",
    "uses": [
      "0P799D6AMN",
      "0P24YW565F",
      "0KWXYZPJKN",
      "0W809K6AW0"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/RATQ474V7S/",
  "links": [
    {
      "field": "proves",
      "target": "0SPYAGE7T0"
    },
    {
      "field": "uses",
      "target": "0P799D6AMN"
    },
    {
      "field": "uses",
      "target": "0P24YW565F"
    },
    {
      "field": "uses",
      "target": "0KWXYZPJKN"
    },
    {
      "field": "uses",
      "target": "0W809K6AW0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RATQ474V7S.md",
  "msc": [
    "15A63",
    "15A21"
  ],
  "name": "Proof of the change of basis rule for a bilinear form",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RAT/RATQ474V7S.md",
  "status": null,
  "tag": "RATQ474V7S",
  "uri": "m2://RATQ474V7S",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-change-of-basis-rule-for-a-bilinear-form/",
  "writing": {
    "tag": "RATQ474V7S",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}