# Question 16416: Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set [Return to catalogue page 83](https://ocme.evemisslab.com/en/questions/page/83/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r01-vector-subspace-foundations/#q-20261014-advance-20261101-r01-vector-subspace-0091) 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. R0031-16416 · q-20261014-advance-20261101-r01-vector-subspace-0091 ## Original question and givens In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 2, 3]. 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, 2, 3]. 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 · calculationBefore affine nonempty particular y coordinate := div(3, 2) After affine nonempty particular y coordinate = 3/2 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore affine particular first term := mul(0, 0) After affine particular first term = 0 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore affine particular second term := mul(2, 3/2) After affine particular second term = 3 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore affine particular equation value := sum(0, 3) After affine particular equation value = 3 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore nonzero original normal=[0, 2];nonzero right side=3;particular real solution=[0, 3/2];checked equation=3 After nonempty affine line;zero gives0 rather than nonzero right side;affine_line_not_through_zero;subspace=No Conditions: none The nonzero normal gives a line,and the checked real solution rules out emptiness.But the zero vector has left side0 rather than nonzero c,so this affine line omits zero and is not a subspace.Do not silently translate it into a homogeneous line. Step identifiers and dependenciesRule: subspace_affine_nonempty_but_zero_missing · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute affine particular equation value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute affine particular second term with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute affine particular first term with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute affine nonempty particular y coordinate with exact rational arithmetic, not rounded decimals. ## Result ([0, 2, 3], affine_line_not_through_zero, No, affine_line_not_through_zero) 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: fcde30a46a4566f43d21229e97c02e3dde6d17761ce0f3e90f7999367395b59f - candidate record SHA-256: 8dae001a836fdb88456a985611daa96753cb5be94ad1902b8b6924ffb78e3e97 - 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-0091.json)