# Question 18667: Uniqueness in a Sum Does Not Yet Give Ambient Coverage [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-r01-direct-sum-foundations/#q-20261014-advance-20261105-r01-direct-sum-0017) 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. R0036-18667 · q-20261014-advance-20261105-r01-direct-sum-0017 ## Original question and givens Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 17]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum). - G001 — Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 17]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum). ## 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-direct-sum](https://ocme.evemisslab.com/en/mko/mko-direct-sum/) ## Step-by-step derivation S001 · impliesBefore source operation=directsum-internal-audit;original snapshot and mathematics retained After explicit ordered source output fields=[dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum] 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 directsum U:complete ordered matrix=[[1], [0], [0]];rows=3;columns=1 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S001 S003 · calculationBefore directsum U pivot0 column0 normalized entry0 := div(1, 1) After directsum U pivot0 column0 normalized entry0 = 1 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 · impliesBefore directsum U:pivot row=0;column=0;nonzero pivot=1;old row=[1] After normalized full row=[1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S003 S005 · impliesBefore directsum U:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0] After rank=1;pivot columns independent and all original column relations preserved;zero rows retained Conditions: none Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S004 S006 · impliesBefore directsum U:complete ORIGINAL independent columns=[[1, 0, 0]] After declared subspace dimension=1;basis of its full real span;empty basis means zero subspace Conditions: none The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S005 S007 · impliesBefore directsum W:complete ordered matrix=[[0], [1], [17]];rows=3;columns=1 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S006 S008 · impliesBefore directsum W:matrix=[[0], [1], [17]];selected row=1 After swap rows0,1;matrix=[[1], [0], [17]] Conditions: none Swap entire rows invertibly while retaining original column/coefficient labels. Step identifiers and dependenciesRule: finite_linear_invertible_row_swap · Uses: G001, S007 S009 · calculationBefore directsum W pivot0 column0 normalized entry0 := div(1, 1) After directsum W pivot0 column0 normalized entry0 = 1 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 · impliesBefore directsum W:pivot row=0;column=0;nonzero pivot=1;old row=[1] After normalized full row=[1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S009 S011 · calculationBefore directsum W eliminate row2 pivot0 entry0 product := mul(17, 1) After directsum W eliminate row2 pivot0 entry0 product = 17 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 directsum W eliminate row2 pivot0 entry0 difference := sub(17, 17) After directsum W eliminate row2 pivot0 entry0 difference = 0 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 · impliesBefore directsum W:target row=2;old=[17];factor=17;pivot row=[1] After new full row=[0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S012 S014 · impliesBefore directsum W:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0] After rank=1;pivot columns independent and all original column relations preserved;zero rows retained Conditions: none Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S013 S015 · impliesBefore directsum W:complete ORIGINAL independent columns=[[0, 1, 17]] After declared subspace dimension=1;basis of its full real span;empty basis means zero subspace Conditions: none The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S014 S016 · impliesBefore directsum ordered U then W:complete ordered matrix=[[1, 0], [0, 1], [0, 17]];rows=3;columns=2 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S015 S017 · calculationBefore directsum ordered U then W pivot0 column0 normalized entry0 := div(1, 1) After directsum ordered U then W pivot0 column0 normalized entry0 = 1 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 directsum ordered U then W pivot0 column0 normalized entry1 := div(0, 1) After directsum ordered U then W pivot0 column0 normalized entry1 = 0 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 · impliesBefore directsum ordered U then W:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0] After normalized full row=[1, 0] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S018 S020 · calculationBefore directsum ordered U then W pivot1 column1 normalized entry0 := div(0, 1) After directsum ordered U then W pivot1 column1 normalized entry0 = 0 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 S021 · calculationBefore directsum ordered U then W pivot1 column1 normalized entry1 := div(1, 1) After directsum ordered U then W pivot1 column1 normalized entry1 = 1 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, S020 S022 · impliesBefore directsum ordered U then W:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S021 S023 · calculationBefore directsum ordered U then W eliminate row2 pivot1 entry0 product := mul(17, 0) After directsum ordered U then W eliminate row2 pivot1 entry0 product = 0 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, S022 S024 · calculationBefore directsum ordered U then W eliminate row2 pivot1 entry0 difference := sub(0, 0) After directsum ordered U then W eliminate row2 pivot1 entry0 difference = 0 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, S023 S025 · calculationBefore directsum ordered U then W eliminate row2 pivot1 entry1 product := mul(17, 1) After directsum ordered U then W eliminate row2 pivot1 entry1 product = 17 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, S024 S026 · calculationBefore directsum ordered U then W eliminate row2 pivot1 entry1 difference := sub(17, 17) After directsum ordered U then W eliminate row2 pivot1 entry1 difference = 0 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, S025 S027 · impliesBefore directsum ordered U then W:target row=2;old=[0, 17];factor=17;pivot row=[0, 1] After new full row=[0, 0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S026 S028 · impliesBefore directsum ordered U then W:complete exact reduced matrix=[[1, 0], [0, 1], [0, 0]];pivot columns=[0, 1] After rank=2;pivot columns independent and all original column relations preserved;zero rows retained Conditions: none Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S027 S029 · impliesBefore directsum U plus W:pivot columns=[0, 1];ALL free columns=[];full relation family=[] After EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=0 Conditions: none Invertible row operations preserve the original kernel.Every zero combination's free coordinates are its unique coefficients in this family,and pivot equations determine the rest.Conversely all these combinations map to zero.The free-coordinate identity proves independence,not merely sampled relations. Step identifiers and dependenciesRule: directsum_full_relation_space_both_directions · Uses: G001, S028 S030 · calculationBefore directsum U and W basis count sum := sum(1, 1) After directsum U and W basis count sum = 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, S029 S031 · calculationBefore directsum actual intersection dimension := sub(2, 2) After directsum actual intersection dimension = 0 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, S030 S032 · impliesBefore ORIGINAL U basis=[[1, 0, 0]];W basis=[[0, 1, 17]];combined rank=2;full zero-combination relations=[];common vectors=[] After kernel of sum map consists of(u,-u) for u in U intersection W;intersection dimension=0;unique decomposition INSIDE U+W=Yes;coverage of R^3=No Conditions: none The sum map surjects onto U+W,and its kernel is linearly isomorphic to U∩W via v↦(v,−v).The complete kernel basis gives the intersection dimension.Decomposition differences lie in this kernel,so zero intersection gives uniqueness within the sum.Full ambient directness additionally requires rank equal to ambient dimension,not one successfully reconstructed target. Step identifiers and dependenciesRule: directsum_unique_in_sum_and_ambient_coverage_separate · Uses: G001, S031 ## Reverse checks - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute directsum actual intersection dimension with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute directsum U and W basis count sum with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute directsum ordered U then W pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute directsum ordered U then W pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute directsum ordered U then W pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute directsum ordered U then W pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute directsum W eliminate row2 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute directsum W eliminate row2 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - 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 directsum W pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute directsum U pivot0 column0 normalized entry0 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 (1, 1, 2, 0, Yes, No, No) 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: fafebce65796bb8647296c0c621ee285d5dffc8d7ebb05faf1a3506681de74fe - candidate record SHA-256: 1cfdc86320789aaec5d2e39912aa0a3d8ba98daef24991fd92708a101e859a38 - 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-r01-direct-sum-0017.json)