# Question 18768: A Covector Evaluates a Vector but Is Not That Vector [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-0018) 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. R0036-18768 · q-20261014-advance-20261105-r02-dual-vector-space-0018 ## Original question and givens Use row coefficients=[18, -1] and primal vector=[2, 20] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary). - G001 — Use row coefficients=[18, -1] and primal vector=[2, 20] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary). ## 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-functional-evaluation;original snapshot and mathematics retained After explicit ordered source output fields=[coefficients, vector, value, boundary] 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=[18, -1];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 functional original evaluation coordinate0 product := mul(18, 2) After dual functional original evaluation coordinate0 product = 36 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 functional original evaluation coordinate1 product := mul(-1, 20) After dual functional original evaluation coordinate1 product = -20 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 functional original evaluation complete sum := sum(36, -20) After dual functional original evaluation complete sum = 16 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 ## Reverse checks - S005 · premise_and_result_audit — Recompute dual functional original evaluation complete sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute dual functional original evaluation coordinate1 product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute dual functional original evaluation coordinate0 product 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 ([18, -1], [2, 20], 16, linear_functional_not_primal_vector) 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: 487cc41b9d07c164f4e235b088225d502de075759247710d96ec394e6c4fa35a - candidate record SHA-256: 9aaae3b84a20bc5624f82eef5d386c660a30c26050c4e250c70fb5d7cd55bd05 - 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-0018.json)