Question 18779: One Row Formula Supplies a Symbolic Linearity Identity

Original question and givens

Use covector row=[4, -2], u=[1, 5], v=[6, -1], alpha=4, 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=[4, -2], u=[1, 5], v=[6, -1], alpha=4, 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=[4, -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(4, 1)

dual linear combination coordinate0 alpha u = 4

dual linear combination coordinate0 beta v := mul(-2, 6)

dual linear combination coordinate0 beta v = -12

dual linear combination coordinate0 sum := sum(4, -12)

dual linear combination coordinate0 sum = -8

dual linear combination coordinate1 alpha u := mul(4, 5)

dual linear combination coordinate1 alpha u = 20

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

dual linear combination coordinate1 beta v = 2

dual linear combination coordinate1 sum := sum(20, 2)

dual linear combination coordinate1 sum = 22

dual linearity left evaluation coordinate0 product := mul(4, -8)

dual linearity left evaluation coordinate0 product = -32

dual linearity left evaluation coordinate1 product := mul(-2, 22)

dual linearity left evaluation coordinate1 product = -44

dual linearity left evaluation complete sum := sum(-32, -44)

dual linearity left evaluation complete sum = -76

dual linearity phi u coordinate0 product := mul(4, 1)

dual linearity phi u coordinate0 product = 4

dual linearity phi u coordinate1 product := mul(-2, 5)

dual linearity phi u coordinate1 product = -10

dual linearity phi u complete sum := sum(4, -10)

dual linearity phi u complete sum = -6

dual linearity phi v coordinate0 product := mul(4, 6)

dual linearity phi v coordinate0 product = 24

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

dual linearity phi v coordinate1 product = 2

dual linearity phi v complete sum := sum(24, 2)

dual linearity phi v complete sum = 26

dual linearity alpha times phi u := mul(4, -6)

dual linearity alpha times phi u = -24

dual linearity beta times phi v := mul(-2, 26)

dual linearity beta times phi v = -52

dual linearity right complete sum := sum(-24, -52)

dual linearity right complete sum = -76

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

(-76, -76, 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.