# Question 16480: A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

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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.

R0031-16480 · q-20261014-advance-20261101-r01-vector-subspace-0030



## Original question and givens

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).


- G001 — Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).





## 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

same usual real R2;fixed normal=[5, -5];U consists of ALL x with(5)*x0+(-5)*x1=0

After

zero vector:( 5)*0+(-5)*0=0;zero belongs to U

Conditions: none

The equation defines the whole set,not a finite sample.First substitute the zero vector;all later vectors and scalars stay in the same usual real vector space.

Step identifiers and dependenciesRule: subspace_homogeneous_zero_vector_universal_parent · Uses: G001





S002 · impliesBefore

ANY u,v in this U;(5)*u0+(-5)*u1=0 and(5)*v0+(-5)*v1=0

After

(5)*(u0+v0)+(-5)*(u1+v1)=((5)*u0+(-5)*u1)+((5)*v0+(-5)*v1)=0+0=0;u+v in U

Conditions: none

Distribute and regroup into the two equations already equal to zero.This proves closure for any two solutions,not just selected points.

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





S003 · impliesBefore

ANY real scalar r and ANY u in U;(5)*u0+(-5)*u1=0

After

(5)*(r*u0)+(-5)*(r*u1)=r*((5)*u0+(-5)*u1)=r*0=0;r*u in U

Conditions: none

Commute the fixed coefficients with an arbitrary real r and factor it out.The result is zero,including zero,negative and irrational scalars.Finitely many integer tests are not a substitute for every-real-scalar closure.

Step identifiers and dependenciesRule: subspace_every_real_scalar_not_finite_sample_scalars · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

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: 21f71c30af82a00f01e28d2273d73aeef1fff52097d11c522b4d1d3da204d0ed
- candidate record SHA-256: 1f73c6327cfad36608b57665a5994989ed5273637ee26274b48dda5f95392cef
- 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-0030.json)