# Question 16421: Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

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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.

R0031-16421 · q-20261014-advance-20261101-r01-vector-subspace-0096



## Original question and givens

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.


- G001 — In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.





## 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-vector-subspace](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/)





## Step-by-step derivation

S001 · impliesBefore

original equation coefficients=[0, 0, 8];left side is identically0

After

equation0=8;classification=empty_set;subspace=No

Conditions: none

When both coefficients vanish,split first:c=0 gives every real vector,while nonzero c gives no solution and omits zero.Neither case is a nondegenerate line.

Step identifiers and dependenciesRule: subspace_zero_normal_whole_or_empty_not_line · Uses: G001









## Reverse checks


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





## Result

([0, 0, 8], empty_set, No, empty_set)

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: 7e8c9efb6d33875f476f93c352dbf4fa9aa312141f4da95dade33070e7a4417d
- candidate record SHA-256: 444202076f82d3a36007247c1894cf772bf1ce086a5f256179cfae1274701bbd
- semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a
- question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r01-vector-subspace-0096.json)