Universal closure of a line through zero

Goal and scope

U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not.

A homogeneous line, zero space, whole R2 and explicit counterexamples; no sampled positive proof.

Givens and symbols

• calc
• subspace_def

• G1 — V=R2, U={(x,y):y-2x=0}, with usual vector addition and all real scalars.

• U — homogeneous line
• \lambda — arbitrary real scalar

Rules and sources

• calc · prerequisite — Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations.
• mko_definition · cited — This slice handles the zero space, the whole space, lines through the origin and explicit finite counterexamples in R². Positive cases use universal algebraic certificates from homogeneous linear equations or spans. Finitely many successful samples do not prove a subspace. · mko-vector-subspace
• subspace_def · prerequisite — A subset U of a real vector space V is a subspace under restricted operations if it contains zero and is closed under addition and every real scalar; the remaining vector laws are inherited from V.

Main derivation

(x,y)\in U

y=2x\iff(x,y)=x(1,2)

0\in\mathbb R

(0,0)=0(1,2)\in U

u=a(1,2),\ v=b(1,2),\quad a,b\in\mathbb R

u+v=(a+b)(1,2)\in U

u=a(1,2),\quad\lambda\in\mathbb R

\lambda u=(\lambda a)(1,2)\in U

0\in U,\quad U+U\subseteq U,\quad\mathbb R U\subseteq U

U\le\mathbb R^2

Z=\{(0,0)\},\quad V=\mathbb R^2

Z+Z=Z,\ \lambda Z\subseteq Z;\quad V+V\subseteq V,\ \lambda V\subseteq V

A=\{(x,y):y=2x+1\}

(0,0)\notin A

B=\{(x,0):x\in\mathbb R\}\cup\{(0,y):y\in\mathbb R\}

(1,0),(0,1)\in B,\quad(1,1)\notin B

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S03 — One explicit counterexample refutes universal closure; finitely many positive samples cannot prove it.
• S07 · premise_audit · G1, S02 — (0,1) lies in A but zero times it does not, also directly refuting scalar closure.
• S06 · premise_audit · G1 — Both contain zero, and addition and all scalars are checked separately.
• S05 · premise_audit · G1, S02, S03, S04 — The certificate uses the usual operations and cannot be reused unchanged for different ones.
• S04 · premise_audit · G1, S01 — Lambda is neither bounded nor restricted to integers; checking only 2 is insufficient.
• S03 · premise_audit · G1, S01 — a and b are arbitrary; two numerical examples would not replace the proof.
• S02 · premise_audit · G1, S01 — Containing zero is necessary but does not yet prove closure.
• S01 · inverse_equivalence · G1 — Substitution gives 2x-2x=0, verifying both directions.

Error controls and negative controls

All real scalars

Closure must hold for every real lambda, not merely integers.

Homogeneity

y-2x=0 contains zero; replacing zero by 1 gives a different affine set.

• The line y=2x+1 is a subspace. → reject — It omits (0,0).
• Containing zero and closure under integer multiples suffice. → reject — Z2 is closed under addition and integer multiples, but (1/2)(1,0) is outside it.
• A union of two subspaces is a subspace. → reject — The union of axes omits (1,0)+(0,1)=(1,1).

Result and limitations

U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 866398e5461531188f4557b3cabd302424ab3b6911d677de331ab220569bbdb6
• original source SHA-256: 866398e5461531188f4557b3cabd302424ab3b6911d677de331ab220569bbdb6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)