# Question 13910: One Passing Vector Does Not Certify a Uniform Bound [Return to catalogue page 70](https://ocme.evemisslab.com/en/questions/page/70/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations/#q-20261014-advance-20261028-r05-bounded-linear-operator-0060) 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. R0026-13910 · q-20261014-advance-20261028-r05-bounded-linear-operator-0060 ## Original question and givens Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-4, 2, -1], proposed C=5. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only). - G001 — Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-4, 2, -1], proposed C=5. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only). ## 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-bounded-linear-operator](https://ocme.evemisslab.com/en/mko/mko-bounded-linear-operator/) ## Step-by-step derivation S001 · impliesBefore provided input image:matrix=[[-3, -1, 0], [2, 3, 0]];input=[-4, 2, -1] After input dimension=3;output dimension=2;rows dot input Conditions: none Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped. Step identifiers and dependenciesRule: matrix_row_column_roles · Uses: G001 S002 · calculationBefore provided input image row0 column0 product := mul(-3, -4) After provided input image row0 column0 product = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore provided input image row0 column1 product := mul(-1, 2) After provided input image row0 column1 product = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore provided input image row0 column2 product := mul(0, -1) After provided input image row0 column2 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore provided input image row0 sum := sum(12, -2, 0) After provided input image row0 sum = 10 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore provided input image row1 column0 product := mul(2, -4) After provided input image row1 column0 product = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore provided input image row1 column1 product := mul(3, 2) After provided input image row1 column1 product = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore provided input image row1 column2 product := mul(0, -1) After provided input image row1 column2 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore provided input image row1 sum := sum(-8, 6, 0) After provided input image row1 sum = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore provided input:signed vector=[-4, 2, -1];norm=linfinity After coordinate absolute values=[4, 2, 1] Conditions: none Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds. Step identifiers and dependenciesRule: operator_named_norm_original_signs · Uses: G001, S009 S011 · impliesBefore provided input:all absolute coordinates=[4, 2, 1] After maximum=4;first attaining coordinate=0 Conditions: none The finite nonempty maximum must bound every value and be attained,not be the largest signed coordinate. Step identifiers and dependenciesRule: operator_linfinity_full_maximum · Uses: G001, S010 S012 · impliesBefore provided output:signed vector=[10, -2];norm=linfinity After coordinate absolute values=[10, 2] Conditions: none Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds. Step identifiers and dependenciesRule: operator_named_norm_original_signs · Uses: G001, S011 S013 · impliesBefore provided output:all absolute coordinates=[10, 2] After maximum=10;first attaining coordinate=0 Conditions: none The finite nonempty maximum must bound every value and be attained,not be the largest signed coordinate. Step identifiers and dependenciesRule: operator_linfinity_full_maximum · Uses: G001, S012 S014 · calculationBefore proposed C times input norm := mul(5, 4) After proposed C times input norm = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore provided input=[-4, 2, -1];output norm=10;C=5;input norm=4;rhs=20 After this instance passes=Yes;instance_only Conditions: none Compare only this input.A zero input can pass even for a nonzero matrix with C=0;that is not a bound for every vector or an exact norm. Step identifiers and dependenciesRule: operator_single_input_not_uniform_exact_norm · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute proposed C times input norm with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute provided input image row1 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute provided input image row1 column2 product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute provided input image row1 column1 product with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute provided input image row1 column0 product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute provided input image row0 sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute provided input image row0 column2 product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute provided input image row0 column1 product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute provided input image row0 column0 product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (linfinity, [-4, 2, -1], [10, -2], 4, 10, 5, 20, Yes, instance_only) 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: 5a91c4eb12ded91d9c5a164ea06f8895d236cc8d965ff22495e0e8350cd0348d - candidate record SHA-256: 7ae63d4da87f4981eb5a51a5efe16692f162105efd05d09fc0eb3d631b2f54e4 - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r05-bounded-linear-operator-0060.json)