# Unique Decompositions, Ambient Coverage and Multi-Summand Independence

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r01-direct-sum-0100/)

Direct sum of subspaces · qf-mod36-directsum-external-coordinate · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r01-direct-sum-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r01-direct-sum-foundations/validation-report.json)