{
  "name": "Linear algebra over GF(2)",
  "version": "1.0.0",
  "area": {
    "name": "Linear algebra over GF(2)",
    "slug": "linear-algebra",
    "group": "Mathematical structures",
    "kind": "linear",
    "summary": "Perform elimination and solve parity equations in a field where addition is XOR, keeping the entire kernel inspectable.",
    "definitions": [
      {
        "term": "GF(2)",
        "definition": "The field with elements 0 and 1; addition and subtraction are XOR."
      },
      {
        "term": "Rank",
        "definition": "The number of pivot columns after elimination."
      },
      {
        "term": "Nullspace",
        "definition": "Every vector x with Ax=0, under arithmetic modulo two."
      }
    ],
    "methodology": [
      "Reduce the binary matrix by row swapping and XOR elimination.",
      "Identify pivot columns and compute rank and nullity.",
      "Enumerate every binary vector of the declared column dimension.",
      "Compare the full kernel list and verify its size against rank-nullity."
    ],
    "complexity": "For m rows and n columns, elimination is polynomial; full kernel enumeration checks 2ⁿ vectors.",
    "common_error": "Ordinary real-number arithmetic gives different answers. A few null vectors need not span the kernel.",
    "next_question": "Add row-operation certificates, nullspace bases, and inconsistency witnesses.",
    "references": [
      "https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/echelon_matrix.html"
    ]
  },
  "records": [
    {
      "id": "KL-FCS-064",
      "version": "1.0.0",
      "domain": "Linear algebra over GF(2)",
      "kind": "linear",
      "title": "Dependent parity equations",
      "problem": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
      "specification": {
        "matrix": [
          [
            1,
            1,
            0,
            1
          ],
          [
            0,
            1,
            1,
            0
          ],
          [
            1,
            0,
            1,
            1
          ]
        ],
        "field": "GF(2); column vectors; all dot products modulo two."
      },
      "claim": {
        "rank": 2,
        "nullity": 2
      },
      "witness": {
        "rref": [
          [
            1,
            0,
            1,
            1
          ],
          [
            0,
            1,
            1,
            0
          ],
          [
            0,
            0,
            0,
            0
          ]
        ],
        "kernel": [
          [
            0,
            0,
            0,
            0
          ],
          [
            0,
            1,
            1,
            1
          ],
          [
            1,
            0,
            0,
            1
          ],
          [
            1,
            1,
            1,
            0
          ]
        ]
      },
      "verification_scope": "Elimination + complete kernel · 16 vectors",
      "explanation": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
      "limitations": "Only these exact matrices are certified; the witness is a complete kernel list rather than a scalable basis certificate.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/echelon_matrix.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "linear-algebra",
        "task": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Elimination + complete kernel · 16 vectors",
        "acceptance": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Perform elimination and solve parity equations in a field where addition is XOR, keeping the entire kernel inspectable.",
        "definitions": [
          {
            "term": "GF(2)",
            "definition": "The field with elements 0 and 1; addition and subtraction are XOR."
          },
          {
            "term": "Rank",
            "definition": "The number of pivot columns after elimination."
          },
          {
            "term": "Nullspace",
            "definition": "Every vector x with Ax=0, under arithmetic modulo two."
          }
        ],
        "reasoning": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "worked_example": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
        "complexity": "For m rows and n columns, elimination is polynomial; full kernel enumeration checks 2ⁿ vectors.",
        "common_error": "Ordinary real-number arithmetic gives different answers. A few null vectors need not span the kernel.",
        "further_work": "Add row-operation certificates, nullspace bases, and inconsistency witnesses."
      },
      "related_ids": [
        "KL-FCS-065",
        "KL-FCS-066"
      ]
    },
    {
      "id": "KL-FCS-065",
      "version": "1.0.0",
      "domain": "Linear algebra over GF(2)",
      "kind": "linear",
      "title": "An invertible binary identity matrix",
      "problem": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
      "specification": {
        "matrix": [
          [
            1,
            0,
            0
          ],
          [
            0,
            1,
            0
          ],
          [
            0,
            0,
            1
          ]
        ],
        "field": "GF(2); column vectors; all dot products modulo two."
      },
      "claim": {
        "rank": 3,
        "nullity": 0
      },
      "witness": {
        "rref": [
          [
            1,
            0,
            0
          ],
          [
            0,
            1,
            0
          ],
          [
            0,
            0,
            1
          ]
        ],
        "kernel": [
          [
            0,
            0,
            0
          ]
        ]
      },
      "verification_scope": "Elimination + complete kernel · 8 vectors",
      "explanation": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
      "limitations": "Only these exact matrices are certified; the witness is a complete kernel list rather than a scalable basis certificate.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/echelon_matrix.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "linear-algebra",
        "task": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Elimination + complete kernel · 8 vectors",
        "acceptance": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Perform elimination and solve parity equations in a field where addition is XOR, keeping the entire kernel inspectable.",
        "definitions": [
          {
            "term": "GF(2)",
            "definition": "The field with elements 0 and 1; addition and subtraction are XOR."
          },
          {
            "term": "Rank",
            "definition": "The number of pivot columns after elimination."
          },
          {
            "term": "Nullspace",
            "definition": "Every vector x with Ax=0, under arithmetic modulo two."
          }
        ],
        "reasoning": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "worked_example": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
        "complexity": "For m rows and n columns, elimination is polynomial; full kernel enumeration checks 2ⁿ vectors.",
        "common_error": "Ordinary real-number arithmetic gives different answers. A few null vectors need not span the kernel.",
        "further_work": "Add row-operation certificates, nullspace bases, and inconsistency witnesses."
      },
      "related_ids": [
        "KL-FCS-064",
        "KL-FCS-066"
      ]
    },
    {
      "id": "KL-FCS-066",
      "version": "1.0.0",
      "domain": "Linear algebra over GF(2)",
      "kind": "linear",
      "title": "One equation with four binary variables",
      "problem": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
      "specification": {
        "matrix": [
          [
            1,
            1,
            1,
            1
          ],
          [
            1,
            1,
            1,
            1
          ]
        ],
        "field": "GF(2); column vectors; all dot products modulo two."
      },
      "claim": {
        "rank": 1,
        "nullity": 3
      },
      "witness": {
        "rref": [
          [
            1,
            1,
            1,
            1
          ],
          [
            0,
            0,
            0,
            0
          ]
        ],
        "kernel": [
          [
            0,
            0,
            0,
            0
          ],
          [
            0,
            0,
            1,
            1
          ],
          [
            0,
            1,
            0,
            1
          ],
          [
            0,
            1,
            1,
            0
          ],
          [
            1,
            0,
            0,
            1
          ],
          [
            1,
            0,
            1,
            0
          ],
          [
            1,
            1,
            0,
            0
          ],
          [
            1,
            1,
            1,
            1
          ]
        ]
      },
      "verification_scope": "Elimination + complete kernel · 16 vectors",
      "explanation": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
      "limitations": "Only these exact matrices are certified; the witness is a complete kernel list rather than a scalable basis certificate.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/echelon_matrix.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "linear-algebra",
        "task": "Compute rank, nullity, reduced row-echelon form, and the complete binary kernel.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Elimination + complete kernel · 16 vectors",
        "acceptance": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Perform elimination and solve parity equations in a field where addition is XOR, keeping the entire kernel inspectable.",
        "definitions": [
          {
            "term": "GF(2)",
            "definition": "The field with elements 0 and 1; addition and subtraction are XOR."
          },
          {
            "term": "Rank",
            "definition": "The number of pivot columns after elimination."
          },
          {
            "term": "Nullspace",
            "definition": "Every vector x with Ax=0, under arithmetic modulo two."
          }
        ],
        "reasoning": [
          "Reduce the binary matrix by row swapping and XOR elimination.",
          "Identify pivot columns and compute rank and nullity.",
          "Enumerate every binary vector of the declared column dimension.",
          "Compare the full kernel list and verify its size against rank-nullity."
        ],
        "worked_example": "XOR row operations preserve the solution space. Free columns account for the kernel’s degrees of freedom, and full enumeration checks every binary candidate.",
        "complexity": "For m rows and n columns, elimination is polynomial; full kernel enumeration checks 2ⁿ vectors.",
        "common_error": "Ordinary real-number arithmetic gives different answers. A few null vectors need not span the kernel.",
        "further_work": "Add row-operation certificates, nullspace bases, and inconsistency witnesses."
      },
      "related_ids": [
        "KL-FCS-064",
        "KL-FCS-065"
      ]
    }
  ],
  "verification": [
    {
      "id": "KL-FCS-064",
      "status": "mechanically-checked",
      "check_units": 16,
      "scope": "Elimination + complete kernel · 16 vectors",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-065",
      "status": "mechanically-checked",
      "check_units": 8,
      "scope": "Elimination + complete kernel · 8 vectors",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-066",
      "status": "mechanically-checked",
      "check_units": 16,
      "scope": "Elimination + complete kernel · 16 vectors",
      "review_status": "awaiting-independent-review"
    }
  ]
}