# Question 18794: One Row Formula Supplies a Symbolic Linearity Identity [Return to catalogue page 94](https://ocme.evemisslab.com/en/questions/page/94/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/#q-20261014-advance-20261105-r02-dual-vector-space-0069) 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. R0036-18794 · q-20261014-advance-20261105-r02-dual-vector-space-0069 ## Original question and givens Use covector row=[19, -2], u=[1, 20], v=[21, -1], alpha=19, 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=[19, -2], u=[1, 20], v=[21, -1], alpha=19, 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](https://ocme.evemisslab.com/en/mko/mko-dual-vector-space/) ## Step-by-step derivation S001 · impliesBefore source operation=dual-linearity-certificate;original snapshot and mathematics retained After explicit ordered source output fields=[left, right, equal, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore fixed original covector ROW=[19, -2];coordinate dimension=2 After 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) Conditions: none Finite distributivity for a fixed row holds for arbitrary real u,v,alpha,beta and gives the universal linearity certificate.The subsequent substitution checks this instance,not linearity of an unspecified function from one passing test.The row represents a real-valued function,not the primal vector itself. Step identifiers and dependenciesRule: dual_row_symbolic_universal_linearity_not_one_numeric_test · Uses: G001, S001 S003 · calculationBefore dual linear combination coordinate0 alpha u := mul(19, 1) After dual linear combination coordinate0 alpha u = 19 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore dual linear combination coordinate0 beta v := mul(-2, 21) After dual linear combination coordinate0 beta v = -42 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore dual linear combination coordinate0 sum := sum(19, -42) After dual linear combination coordinate0 sum = -23 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore dual linear combination coordinate1 alpha u := mul(19, 20) After dual linear combination coordinate1 alpha u = 380 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore dual linear combination coordinate1 beta v := mul(-2, -1) After dual linear combination coordinate1 beta v = 2 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore dual linear combination coordinate1 sum := sum(380, 2) After dual linear combination coordinate1 sum = 382 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore dual linearity left evaluation coordinate0 product := mul(19, -23) After dual linearity left evaluation coordinate0 product = -437 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore dual linearity left evaluation coordinate1 product := mul(-2, 382) After dual linearity left evaluation coordinate1 product = -764 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore dual linearity left evaluation complete sum := sum(-437, -764) After dual linearity left evaluation complete sum = -1201 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore dual linearity phi u coordinate0 product := mul(19, 1) After dual linearity phi u coordinate0 product = 19 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore dual linearity phi u coordinate1 product := mul(-2, 20) After dual linearity phi u coordinate1 product = -40 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore dual linearity phi u complete sum := sum(19, -40) After dual linearity phi u complete sum = -21 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore dual linearity phi v coordinate0 product := mul(19, 21) After dual linearity phi v coordinate0 product = 399 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore dual linearity phi v coordinate1 product := mul(-2, -1) After dual linearity phi v coordinate1 product = 2 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore dual linearity phi v complete sum := sum(399, 2) After dual linearity phi v complete sum = 401 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore dual linearity alpha times phi u := mul(19, -21) After dual linearity alpha times phi u = -399 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore dual linearity beta times phi v := mul(-2, 401) After dual linearity beta times phi v = -802 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore dual linearity right complete sum := sum(-399, -802) After dual linearity right complete sum = -1201 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 ## 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 (-1201, -1201, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 810fd46644adb11af92bcb1f193a49dd7d0c35331674dc979bb6d025de35dacd - candidate record SHA-256: 24044278bd5979b495293f716a6f89ca0716ff776db278e04064f0df79977b0e - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r02-dual-vector-space-0069.json)