# Question 25805: Construct the Dual from One Fixed Primal Convention and the Exact Transpose

[Return to catalogue page 130](https://ocme.evemisslab.com/en/questions/page/130/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r04-linear-program-duality-foundations/#q-20261015-advance-20261115-r04-linear-program-duality-0005)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0050-25805 · q-20261015-advance-20261115-r04-linear-program-duality-0005



## Original question and givens

Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).


- G001 — Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).





## 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.


- [mko-linear-program-duality](https://ocme.evemisslab.com/en/mko/mko-linear-program-duality/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [lpdual-construction-qualification]

After

derived = [[constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate]]

Conditions: none

List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [[[1, 0], [0, 1]], [6, 7], [1, 5]]

After

derived = [2, 2, [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes]

Conditions: none

The fixed max/<=/x>=0 convention pairs with min/>=/y>=0.The dual has m variables and n constraints.Transpose every position;either coefficient sign is legal and dimensions do not imply feasibility.

Step identifiers and dependenciesRule: LPdual_fixed_conventions_complete_transpose_and_dimensions · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)

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: cbb42dbfaae73e34e9c2bad7f2499417cd6c9f9971c0d362cb08b1861cc3a43a
- candidate record SHA-256: 175baef1c53e44fbaf8aa3bc13a92ab25a5297fd11c5a18fc85c6dfad161ebde
- semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
- question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r04-linear-program-duality-0005.json)