Question 18791: One Row Formula Supplies a Symbolic Linearity Identity

Original question and givens

Use covector row=[16, -2], u=[1, 17], v=[18, -1], alpha=16, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

• G001 — Use covector row=[16, -2], u=[1, 17], v=[18, -1], alpha=16, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, 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-dual-vector-space

Step-by-step derivation

source operation=dual-linearity-certificate;original snapshot and mathematics retained

explicit ordered source output fields=[left, right, equal, certificate]

fixed original covector ROW=[16, -2];coordinate dimension=2

define phi(x)=sum_j row[j]*x[j] for EVERY real coordinate vector x;finite distributivity gives phi(alpha*u+beta*v)=alpha*phi(u)+beta*phi(v)

dual linear combination coordinate0 alpha u := mul(16, 1)

dual linear combination coordinate0 alpha u = 16

dual linear combination coordinate0 beta v := mul(-2, 18)

dual linear combination coordinate0 beta v = -36

dual linear combination coordinate0 sum := sum(16, -36)

dual linear combination coordinate0 sum = -20

dual linear combination coordinate1 alpha u := mul(16, 17)

dual linear combination coordinate1 alpha u = 272

dual linear combination coordinate1 beta v := mul(-2, -1)

dual linear combination coordinate1 beta v = 2

dual linear combination coordinate1 sum := sum(272, 2)

dual linear combination coordinate1 sum = 274

dual linearity left evaluation coordinate0 product := mul(16, -20)

dual linearity left evaluation coordinate0 product = -320

dual linearity left evaluation coordinate1 product := mul(-2, 274)

dual linearity left evaluation coordinate1 product = -548

dual linearity left evaluation complete sum := sum(-320, -548)

dual linearity left evaluation complete sum = -868

dual linearity phi u coordinate0 product := mul(16, 1)

dual linearity phi u coordinate0 product = 16

dual linearity phi u coordinate1 product := mul(-2, 17)

dual linearity phi u coordinate1 product = -34

dual linearity phi u complete sum := sum(16, -34)

dual linearity phi u complete sum = -18

dual linearity phi v coordinate0 product := mul(16, 18)

dual linearity phi v coordinate0 product = 288

dual linearity phi v coordinate1 product := mul(-2, -1)

dual linearity phi v coordinate1 product = 2

dual linearity phi v complete sum := sum(288, 2)

dual linearity phi v complete sum = 290

dual linearity alpha times phi u := mul(16, -18)

dual linearity alpha times phi u = -288

dual linearity beta times phi v := mul(-2, 290)

dual linearity beta times phi v = -580

dual linearity right complete sum := sum(-288, -580)

dual linearity right complete sum = -868

Reverse checks

• S020 · premise_and_result_audit — Recompute dual linearity right complete sum with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute dual linearity beta times phi v with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute dual linearity alpha times phi u with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute dual linearity phi v complete sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute dual linearity phi v coordinate1 product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute dual linearity phi v coordinate0 product with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute dual linearity phi u complete sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute dual linearity phi u coordinate1 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute dual linearity phi u coordinate0 product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute dual linearity left evaluation complete sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute dual linearity left evaluation coordinate1 product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute dual linearity left evaluation coordinate0 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute dual linear combination coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute dual linear combination coordinate1 beta v with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute dual linear combination coordinate1 alpha u with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute dual linear combination coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute dual linear combination coordinate0 beta v with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute dual linear combination coordinate0 alpha u with exact rational arithmetic, not rounded decimals.
• 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

(-868, -868, Yes, row_covector_gives_universal_linearity_certificate)

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.