# Bilinear Evaluation, Symmetry and Strict Sylvester Inner-Product Qualification

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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-20261028-r04-real-inner-product-space-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.



## 1. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, -3], y=[-5, 12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0001/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 2. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -2], y=[-4, 11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0002/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 3. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -1], y=[-3, 10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [0, -1], [-3, 10], -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0003/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 4. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 0], y=[-2, 9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [1, 0], [-2, 9], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0004/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 5. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 1], y=[-1, 8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0005/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 6. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 2], y=[0, 7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], [-2, 2], [0, 7], 14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0006/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 7. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, 3], y=[1, 6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [-1, 3], [1, 6], 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0007/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 8. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -3], y=[2, 5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[9, 2], [2, 5]], [0, -3], [2, 5], -87)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [0, -3], [2, 5], -87).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0008/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 9. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [1, -2], [3, 4], 16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0009/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 10. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, -1], y=[4, 3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[11, 0], [0, 1]], [2, -1], [4, 3], 85)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [2, -1], [4, 3], 85).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0010/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 11. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 0], y=[5, 2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], [-2, 0], [5, 2], -124).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0011/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 12. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, 1], y=[-5, 1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0012/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 13. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, 2], y=[-4, 0]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0013/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 14. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 3], y=[-3, -1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [1, 3], [-3, -1], -48).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0014/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 15. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, -3], y=[-2, -2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [2, -3], [-2, -2], -50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0015/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 16. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0016/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 17. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -1], y=[0, -4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0017/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 18. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, 0], y=[1, -5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[19, 0], [0, 1]], [0, 0], [1, -5], 0)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], [1, -5], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0018/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 19. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 1], y=[2, -6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[20, 1], [1, 2]], [1, 1], [2, -6], 24)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], [2, -6], 24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0019/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 20. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 2], y=[3, -7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[21, 2], [2, 5]], [2, 2], [3, -7], 40)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], [3, -7], 40).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0020/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 21. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 3], y=[4, -8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0021/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 22. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -3], y=[5, -9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [-1, -3], [5, -9], -88).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0022/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 23. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -2], y=[-5, -10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [0, -2], [-5, -10], 50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0023/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 24. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -1], y=[-4, -11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [1, -1], [-4, -11], -59).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0024/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 25. Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 0], y=[-3, -12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

View answer and explanation([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)

Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0025/)

Real inner-product space · qf-mod28-inner-evaluate · candidate





## 26. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[2, -1], [0, 2]], [-1, 0], No)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [0, 2]], [-1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0026/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 27. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[3, 0], [0, 1]], [0, 0], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0027/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 28. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[4, 1], [1, 2]], [1, 1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [1, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0028/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 29. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[5, 2], [2, 5]], [2, 2], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [2, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0029/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 30. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[6, -1], [-1, 2]], [-1, -1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [-1, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0030/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 31. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[7, 0], [1, 1]], [0, 1], No)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0031/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 32. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[8, 1], [1, 2]], [1, 1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [1, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0032/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 33. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[9, 2], [2, 5]], [2, 2], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [2, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0033/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 34. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[10, -1], [-1, 2]], [-1, -1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [-1, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0034/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 35. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[11, 0], [0, 1]], [0, 0], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0035/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 36. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[12, 1], [2, 2]], [1, 2], No)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], [1, 2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0036/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 37. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[13, 2], [2, 5]], [2, 2], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [2, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0037/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 38. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[14, -1], [-1, 2]], [-1, -1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [-1, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0038/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 39. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[15, 0], [0, 1]], [0, 0], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0039/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 40. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[16, 1], [1, 2]], [1, 1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [1, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0040/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 41. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[17, 2], [3, 5]], [2, 3], No)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], [2, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0041/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 42. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[18, -1], [-1, 2]], [-1, -1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0042/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 43. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[19, 0], [0, 1]], [0, 0], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0043/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 44. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[20, 1], [1, 2]], [1, 1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0044/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 45. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[21, 2], [2, 5]], [2, 2], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0045/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 46. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[22, -1], [0, 2]], [-1, 0], No)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], [-1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0046/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 47. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[23, 0], [0, 1]], [0, 0], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0047/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 48. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[24, 1], [1, 2]], [1, 1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [1, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0048/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 49. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[25, 2], [2, 5]], [2, 2], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [2, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0049/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 50. Symmetry Alone Does Not Establish an Inner Product

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

View answer and explanation([[26, -1], [-1, 2]], [-1, -1], Yes)

Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [-1, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0050/)

Real inner-product space · qf-mod28-inner-symmetry · candidate





## 51. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0051/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 52. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0052/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 53. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0053/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 54. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0054/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 55. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0055/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 56. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0056/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 57. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0057/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 58. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0058/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 59. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0059/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 60. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0060/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 61. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0061/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 62. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0062/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 63. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0063/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 64. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0064/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 65. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0065/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 66. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0066/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 67. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0067/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 68. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0068/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 69. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0069/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 70. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0070/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 71. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0071/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 72. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0072/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 73. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0073/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 74. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0074/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 75. A Symmetric Sylvester Certificate, Not Positive Sample Values

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.

View answer and explanation([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate)

Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0075/)

Real inner-product space · qf-mod28-inner-positive-definite · candidate





## 76. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0076/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 77. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0077/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 78. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0078/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 79. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0079/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 80. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0080/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 81. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0081/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 82. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0082/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 83. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0083/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 84. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0084/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 85. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0085/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 86. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0086/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 87. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0087/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 88. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0088/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 89. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0089/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 90. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0090/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 91. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0091/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 92. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0092/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 93. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0093/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 94. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0094/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 95. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0095/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 96. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0096/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 97. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0097/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 98. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0098/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 99. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0099/)

Real inner-product space · qf-mod28-inner-qualification · candidate





## 100. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.

View answer and explanation([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)

The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0100/)

Real inner-product space · qf-mod28-inner-qualification · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations/validation-report.json)