Unique Decompositions, Ambient Coverage and Multi-Summand Independence
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batch-20261014-advance-20261105-r01-direct-sum-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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Field-order and source-binding data
1. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 1]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
2. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 2]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
3. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 3]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
4. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 4]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
5. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 5]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
6. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 6]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
7. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 7]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
8. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 8]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
9. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 9]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
10. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 10]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
11. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 11]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
12. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 12]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
13. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 13]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
14. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 14]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
15. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 15]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
16. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 16]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
17. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
18. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 18]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
19. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
20. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 20]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
21. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 21]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
22. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 22]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
23. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 23]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
24. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
25. Uniqueness in a Sum Does Not Yet Give Ambient Coverage
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 25]] 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).
View answer and explanation
(1, 1, 2, 0, Yes, No, No)
Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-internal-audit · candidate
26. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 1]] 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. Target vector=[1, 2, 2], coefficients_u=[1], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([1, 0, 0], [0, 2, 2], [1, 2, 2], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([1, 0, 0], [0, 2, 2], [1, 2, 2], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
27. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 2]] 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. Target vector=[2, 2, 4], coefficients_u=[2], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([2, 0, 0], [0, 2, 4], [2, 2, 4], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([2, 0, 0], [0, 2, 4], [2, 2, 4], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
28. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 3]] 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. Target vector=[3, 2, 6], coefficients_u=[3], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([3, 0, 0], [0, 2, 6], [3, 2, 6], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([3, 0, 0], [0, 2, 6], [3, 2, 6], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
29. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 4]] 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. Target vector=[4, 2, 8], coefficients_u=[4], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([4, 0, 0], [0, 2, 8], [4, 2, 8], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([4, 0, 0], [0, 2, 8], [4, 2, 8], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
30. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 5]] 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. Target vector=[5, 2, 10], coefficients_u=[5], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([5, 0, 0], [0, 2, 10], [5, 2, 10], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([5, 0, 0], [0, 2, 10], [5, 2, 10], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
31. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 6]] 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. Target vector=[6, 2, 12], coefficients_u=[6], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([6, 0, 0], [0, 2, 12], [6, 2, 12], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([6, 0, 0], [0, 2, 12], [6, 2, 12], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
32. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 7]] 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. Target vector=[7, 2, 14], coefficients_u=[7], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([7, 0, 0], [0, 2, 14], [7, 2, 14], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([7, 0, 0], [0, 2, 14], [7, 2, 14], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
33. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 8]] 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. Target vector=[8, 2, 16], coefficients_u=[8], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([8, 0, 0], [0, 2, 16], [8, 2, 16], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([8, 0, 0], [0, 2, 16], [8, 2, 16], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
34. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 9]] 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. Target vector=[9, 2, 18], coefficients_u=[9], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([9, 0, 0], [0, 2, 18], [9, 2, 18], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([9, 0, 0], [0, 2, 18], [9, 2, 18], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
35. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 10]] 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. Target vector=[10, 2, 20], coefficients_u=[10], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([10, 0, 0], [0, 2, 20], [10, 2, 20], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([10, 0, 0], [0, 2, 20], [10, 2, 20], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
36. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 11]] 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. Target vector=[11, 2, 22], coefficients_u=[11], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([11, 0, 0], [0, 2, 22], [11, 2, 22], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([11, 0, 0], [0, 2, 22], [11, 2, 22], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
37. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 12]] 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. Target vector=[12, 2, 24], coefficients_u=[12], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([12, 0, 0], [0, 2, 24], [12, 2, 24], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([12, 0, 0], [0, 2, 24], [12, 2, 24], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
38. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 13]] 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. Target vector=[13, 2, 26], coefficients_u=[13], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([13, 0, 0], [0, 2, 26], [13, 2, 26], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([13, 0, 0], [0, 2, 26], [13, 2, 26], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
39. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 14]] 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. Target vector=[14, 2, 28], coefficients_u=[14], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([14, 0, 0], [0, 2, 28], [14, 2, 28], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([14, 0, 0], [0, 2, 28], [14, 2, 28], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
40. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 15]] 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. Target vector=[15, 2, 30], coefficients_u=[15], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([15, 0, 0], [0, 2, 30], [15, 2, 30], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([15, 0, 0], [0, 2, 30], [15, 2, 30], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
41. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 16]] 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. Target vector=[16, 2, 32], coefficients_u=[16], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([16, 0, 0], [0, 2, 32], [16, 2, 32], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([16, 0, 0], [0, 2, 32], [16, 2, 32], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
42. Check This Decomposition Separately from a Whole-Space Direct Sum
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. Target vector=[17, 2, 34], coefficients_u=[17], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([17, 0, 0], [0, 2, 34], [17, 2, 34], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([17, 0, 0], [0, 2, 34], [17, 2, 34], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
43. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 18]] 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. Target vector=[18, 2, 36], coefficients_u=[18], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([18, 0, 0], [0, 2, 36], [18, 2, 36], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([18, 0, 0], [0, 2, 36], [18, 2, 36], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
44. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] 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. Target vector=[19, 2, 38], coefficients_u=[19], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([19, 0, 0], [0, 2, 38], [19, 2, 38], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([19, 0, 0], [0, 2, 38], [19, 2, 38], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
45. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 20]] 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. Target vector=[20, 2, 40], coefficients_u=[20], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([20, 0, 0], [0, 2, 40], [20, 2, 40], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([20, 0, 0], [0, 2, 40], [20, 2, 40], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
46. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 21]] 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. Target vector=[21, 2, 42], coefficients_u=[21], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([21, 0, 0], [0, 2, 42], [21, 2, 42], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([21, 0, 0], [0, 2, 42], [21, 2, 42], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
47. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 22]] 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. Target vector=[22, 2, 44], coefficients_u=[22], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([22, 0, 0], [0, 2, 44], [22, 2, 44], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([22, 0, 0], [0, 2, 44], [22, 2, 44], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
48. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 23]] 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. Target vector=[23, 2, 46], coefficients_u=[23], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([23, 0, 0], [0, 2, 46], [23, 2, 46], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([23, 0, 0], [0, 2, 46], [23, 2, 46], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
49. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] 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. Target vector=[24, 2, 48], coefficients_u=[24], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
50. Check This Decomposition Separately from a Whole-Space Direct Sum
Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 25]] 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. Target vector=[25, 2, 50], coefficients_u=[25], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).
View answer and explanation
([25, 0, 0], [0, 2, 50], [25, 2, 50], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)
Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([25, 0, 0], [0, 2, 50], [25, 2, 50], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-ambient-decomposition · candidate
51. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 1]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
52. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 2]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
53. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 3]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
54. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 4]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
55. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 5]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
56. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 6]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
57. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 7]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
58. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 8]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
59. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 9]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
60. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 10]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
61. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 11]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
62. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 12]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
63. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 13]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
64. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 14]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
65. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 15]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
66. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 16]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
67. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 17]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
68. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 18]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
69. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 19]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
70. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 20]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
71. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 21]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
72. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 22]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
73. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 23]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
74. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 24]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
75. Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum
Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 25]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).
View answer and explanation
(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)
Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-multi-subspace-global · candidate
76. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[0, 1], right=[2], another pair other_left=[3, 1], other_right=[4], and scalar=1. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
77. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[1, 2], right=[3], another pair other_left=[4, 3], other_right=[5], and scalar=2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
78. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[2, 3], right=[4], another pair other_left=[5, 5], other_right=[6], and scalar=3. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
79. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[3, 4], right=[5], another pair other_left=[6, 7], other_right=[7], and scalar=4. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
80. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[4, 5], right=[6], another pair other_left=[7, 9], other_right=[8], and scalar=5. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
81. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[5, 6], right=[7], another pair other_left=[8, 11], other_right=[9], and scalar=6. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
82. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[6, 7], right=[8], another pair other_left=[9, 13], other_right=[10], and scalar=7. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
83. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[7, 8], right=[9], another pair other_left=[10, 15], other_right=[11], and scalar=8. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
84. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[8, 9], right=[10], another pair other_left=[11, 17], other_right=[12], and scalar=9. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
85. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[9, 10], right=[11], another pair other_left=[12, 19], other_right=[13], and scalar=10. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
86. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[10, 11], right=[12], another pair other_left=[13, 21], other_right=[14], and scalar=11. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
87. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[11, 12], right=[13], another pair other_left=[14, 23], other_right=[15], and scalar=12. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
88. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[12, 13], right=[14], another pair other_left=[15, 25], other_right=[16], and scalar=13. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
89. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[13, 14], right=[15], another pair other_left=[16, 27], other_right=[17], and scalar=14. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
90. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[14, 15], right=[16], another pair other_left=[17, 29], other_right=[18], and scalar=15. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
91. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[15, 16], right=[17], another pair other_left=[18, 31], other_right=[19], and scalar=16. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
92. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[16, 17], right=[18], another pair other_left=[19, 33], other_right=[20], and scalar=17. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
93. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[17, 18], right=[19], another pair other_left=[20, 35], other_right=[21], and scalar=18. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
94. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[18, 19], right=[20], another pair other_left=[21, 37], other_right=[22], and scalar=19. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
95. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[19, 20], right=[21], another pair other_left=[22, 39], other_right=[23], and scalar=20. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
96. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[20, 21], right=[22], another pair other_left=[23, 41], other_right=[24], and scalar=21. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
97. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[21, 22], right=[23], another pair other_left=[24, 43], other_right=[25], and scalar=22. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
98. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[22, 23], right=[24], another pair other_left=[25, 45], other_right=[26], and scalar=23. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
99. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[23, 24], right=[25], another pair other_left=[26, 47], other_right=[27], and scalar=24. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
100. External Direct-Sum Coordinates Live on a Product Carrier
Use the pair (left,right) with left=[24, 25], right=[26], another pair other_left=[27, 49], other_right=[28], and scalar=25. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).
View answer and explanation
([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces)
Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces).
Reviewed step-by-step derivation
Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate
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