OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 27961: Full Row Rank Means the Declared Constraint Gradients Are Independent

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 140

← Question derivation index · Original question batch

R0054-27961 · q-20261015-pulse-20261119-r04-0011

Original question and givens

Use constraint gradients as ordered rows=[[1, 0], [0, 11]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).

  • G001 — Use constraint gradients as ordered rows=[[1, 0], [0, 11]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [lagrange-regularity-audit]

After

derived = [[constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition]]

Conditions: none

List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Lagrange ORIGINAL ORIGINAL normalize row0 col0 := div(1, 1)

After

Lagrange ORIGINAL ORIGINAL normalize row0 col0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Lagrange ORIGINAL ORIGINAL normalize row0 col1 := div(0, 1)

After

Lagrange ORIGINAL ORIGINAL normalize row0 col1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Lagrange ORIGINAL, 0, 0, 0, [[1, 0], [0, 11]], 1]

After

derived = [[[1, 0], [0, 11]]]

Conditions: none

Use the original first-nonzero pivot order and swap rows.Divide only by a nonzero pivot and normalize EVERY column;the inverse operation multiplies back,so row rank is preserved.All unreduced source1e8 intermediates remain qualified.

Step identifiers and dependencies

Rule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S003

S005 · calculation

Before

Lagrange ORIGINAL ORIGINAL normalize row1 col0 := div(0, 11)

After

Lagrange ORIGINAL ORIGINAL normalize row1 col0 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Lagrange ORIGINAL ORIGINAL normalize row1 col1 := div(11, 11)

After

Lagrange ORIGINAL ORIGINAL normalize row1 col1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Lagrange ORIGINAL, 1, 1, 1, [[1, 0], [0, 11]], 11]

After

derived = [[[1, 0], [0, 1]]]

Conditions: none

Use the original first-nonzero pivot order and swap rows.Divide only by a nonzero pivot and normalize EVERY column;the inverse operation multiplies back,so row rank is preserved.All unreduced source1e8 intermediates remain qualified.

Step identifiers and dependencies

Rule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S006

S008 · implies

Before

source roles = [Lagrange ORIGINAL, [[1, 0], [0, 11]]]

After

derived = [[[1, 0], [0, 1]], [[0, 0], [1, 1]], 2]

Conditions: none

Distinct leading1 pivots make the nonzero rows independent;remaining rows are zero and operations are invertible,so pivot count equals original row rank.Retain ORIGINAL constraint rows for J_g/lambda pairing,not RREF substitutes.

Step identifiers and dependencies

Rule: Lagrange_rank_RREF_pivots_equal_row_rank_original_rows_retained · Uses: G001, S007

S009 · implies

Before

source roles = [[[1, 0], [0, 11]]]

After

derived = [2, 2, 2, Yes]

Conditions: none

This qualifies algebraic rank of GIVEN gradient rows only.Without functions/point data it does not independently establish differentiability,feasibility or an extremum.Compare rank with constraint count,not ambient dimension.

Step identifiers and dependencies

Rule: Lagrange_supplied_vector_rank_ONLY_no_unprovided_functions_or_feasibility · Uses: G001, S008

Reverse checks

  • S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S006 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S003 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

([[1, 0], [0, 11]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d96e2b6e776a6fb5f551d5c84c858e1a591672c9033390b365627cddfed590ea
  • candidate record SHA-256: 8f756afb82dc841e624302ed0510e6e17d91a6f06cf642a09f2a740047f073c9
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON