Ordered Matrix Bilinear Values, Separate Slots and Additional Symmetry
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batch-20261014-advance-20261027-r04-real-bilinear-form-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 an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, -3], y=[-5, 12]. Give (matrix,x,y,value).
View answer and explanation
([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
2. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -2], y=[-4, 11]. Give (matrix,x,y,value).
View answer and explanation
([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
3. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -1], y=[-3, 10]. Give (matrix,x,y,value).
View answer and explanation
([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
4. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 0], y=[-2, 9]. Give (matrix,x,y,value).
View answer and explanation
([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
5. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 1], y=[-1, 8]. Give (matrix,x,y,value).
View answer and explanation
([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
6. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 2], y=[0, 7]. Give (matrix,x,y,value).
View answer and explanation
([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
7. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, 3], y=[1, 6]. Give (matrix,x,y,value).
View answer and explanation
([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
8. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -3], y=[2, 5]. Give (matrix,x,y,value).
View answer and explanation
([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
9. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value).
View answer and explanation
([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2], [2, 4]], [1, -2], [3, 4], -64).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
10. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, -1], y=[4, 3]. Give (matrix,x,y,value).
View answer and explanation
([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
11. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 0], y=[5, 2]. Give (matrix,x,y,value).
View answer and explanation
([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
12. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, 1], y=[-5, 1]. Give (matrix,x,y,value).
View answer and explanation
([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
13. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, 2], y=[-4, 0]. Give (matrix,x,y,value).
View answer and explanation
([[0, 2], [0, -1]], [0, 2], [-4, 0], 0)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2], [0, -1]], [0, 2], [-4, 0], 0).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
14. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 3], y=[-3, -1]. Give (matrix,x,y,value).
View answer and explanation
([[1, 3], [5, 0]], [1, 3], [-3, -1], -51)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3], [5, 0]], [1, 3], [-3, -1], -51).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
15. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, -3], y=[-2, -2]. Give (matrix,x,y,value).
View answer and explanation
([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
16. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value).
View answer and explanation
([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
17. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -1], y=[0, -4]. Give (matrix,x,y,value).
View answer and explanation
([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
18. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, 0], y=[1, -5]. Give (matrix,x,y,value).
View answer and explanation
([[5, 0], [3, 4]], [0, 0], [1, -5], 0)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0], [3, 4]], [0, 0], [1, -5], 0).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
19. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 1], y=[2, -6]. Give (matrix,x,y,value).
View answer and explanation
([[6, 1], [-3, -4]], [1, 1], [2, -6], 24)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1], [-3, -4]], [1, 1], [2, -6], 24).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
20. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 2], y=[3, -7]. Give (matrix,x,y,value).
View answer and explanation
([[7, 2], [2, -3]], [2, 2], [3, -7], 68)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2], [2, -3]], [2, 2], [3, -7], 68).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
21. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 3], y=[4, -8]. Give (matrix,x,y,value).
View answer and explanation
([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
22. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -3], y=[5, -9]. Give (matrix,x,y,value).
View answer and explanation
([[9, -3], [1, -1]], [-1, -3], [5, -9], -114)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3], [1, -1]], [-1, -3], [5, -9], -114).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
23. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -2], y=[-5, -10]. Give (matrix,x,y,value).
View answer and explanation
([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
24. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -1], y=[-4, -11]. Give (matrix,x,y,value).
View answer and explanation
([[11, -1], [0, 1]], [1, -1], [-4, -11], -22)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [0, 1]], [1, -1], [-4, -11], -22).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
25. Evaluate an Ordered Real Bilinear Matrix Formula
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 0], y=[-3, -12]. Give (matrix,x,y,value).
View answer and explanation
([[12, 0], [5, 2]], [2, 0], [-3, -12], -72)
First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0], [5, 2]], [2, 0], [-3, -12], -72).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-matrix-evaluate · candidate
26. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[-3, 1], alpha=-2, beta=-3, fixed y=[-5, 12]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-2, -3, [7, 1], 145, 145, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -3, [7, 1], 145, 145, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
27. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-2, 1], alpha=-1, beta=-2, fixed y=[-4, 11]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-1, -2, [3, -1], 99, 99, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -2, [3, -1], 99, 99, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
28. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-1, 1], alpha=0, beta=-1, fixed y=[-3, 10]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(0, -1, [1, -1], 55, 55, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -1, [1, -1], 55, 55, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
29. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[0, 1], alpha=1, beta=0, fixed y=[-2, 9]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(1, 0, [1, 1], 11, 11, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [1, 1], 11, 11, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
30. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[1, 1], alpha=2, beta=1, fixed y=[-1, 8]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(2, 1, [3, 5], 28, 28, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1, [3, 5], 28, 28, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
31. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[2, 1], alpha=-2, beta=2, fixed y=[0, 7]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-2, 2, [2, 6], 70, 70, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [2, 6], 70, 70, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
32. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[3, 1], alpha=-1, beta=3, fixed y=[1, 6]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-1, 3, [8, 4], 156, 156, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, [8, 4], 156, 156, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
33. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-3, 1], alpha=0, beta=-3, fixed y=[2, 5]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(0, -3, [9, -3], -252, -252, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -3, [9, -3], -252, -252, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
34. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-2, 1], alpha=1, beta=-2, fixed y=[3, 4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(1, -2, [5, -1], -122, -122, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [5, -1], -122, -122, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
35. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[-1, 1], alpha=2, beta=-1, fixed y=[4, 3]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(2, -1, [3, 3], -129, -129, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -1, [3, 3], -129, -129, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
36. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[0, 1], alpha=-2, beta=0, fixed y=[5, 2]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-2, 0, [-2, 4], 16, 16, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 0, [-2, 4], 16, 16, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
37. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[1, 1], alpha=-1, beta=1, fixed y=[-5, 1]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-1, 1, [0, 2], 46, 46, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 1, [0, 2], 46, 46, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
38. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[2, 1], alpha=0, beta=2, fixed y=[-4, 0]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(0, 2, [4, 2], 0, 0, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, [4, 2], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
39. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[3, 1], alpha=1, beta=3, fixed y=[-3, -1]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(1, 3, [10, 4], -120, -120, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, [10, 4], -120, -120, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
40. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[-3, 1], alpha=2, beta=-3, fixed y=[-2, -2]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(2, -3, [11, 1], 22, 22, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -3, [11, 1], 22, 22, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
41. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[-2, 1], alpha=-2, beta=-2, fixed y=[-1, -3]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-2, -2, [2, 2], -14, -14, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -2, [2, 2], -14, -14, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
42. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-1, 1], alpha=-1, beta=-1, fixed y=[0, -4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-1, -1, [0, 0], 0, 0, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -1, [0, 0], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
43. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[0, 1], alpha=0, beta=0, fixed y=[1, -5]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(0, 0, [0, 0], 0, 0, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0, 0], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
44. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[1, 1], alpha=1, beta=1, fixed y=[2, -6]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(1, 1, [2, 2], 48, 48, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, [2, 2], 48, 48, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
45. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[2, 1], alpha=2, beta=2, fixed y=[3, -7]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(2, 2, [6, 6], 204, 204, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, [6, 6], 204, 204, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
46. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[3, 1], alpha=-2, beta=3, fixed y=[4, -8]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-2, 3, [7, 7], 56, 56, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, [7, 7], 56, 56, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
47. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-3, 1], alpha=-1, beta=-3, fixed y=[5, -9]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(-1, -3, [8, -2], 548, 548, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, [8, -2], 548, 548, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
48. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-2, 1], alpha=0, beta=-2, fixed y=[-5, -10]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(0, -2, [4, -2], -170, -170, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, [4, -2], -170, -170, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
49. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-1, 1], alpha=1, beta=-1, fixed y=[-4, -11]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(1, -1, [2, 0], -66, -66, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, [2, 0], -66, -66, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
50. Linearity in the First Slot with the Second Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[0, 1], alpha=2, beta=0, fixed y=[-3, -12]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).
View answer and explanation
(2, 0, [2, 4], -228, -228, Yes)
Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [2, 4], -228, -228, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-first-slot-identity · candidate
51. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, -3], u=[1, -2], v=[-3, 1], alpha=-2, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-2, -3, [7, 1], 291, 291, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -3, [7, 1], 291, 291, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
52. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -2], u=[1, -1], v=[-2, 1], alpha=-1, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-1, -2, [3, -1], 25, 25, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -2, [3, -1], 25, 25, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
53. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -1], u=[1, 0], v=[-1, 1], alpha=0, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(0, -1, [1, -1], -7, -7, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -1, [1, -1], -7, -7, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
54. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 0], u=[1, 1], v=[0, 1], alpha=1, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(1, 0, [1, 1], -9, -9, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [1, 1], -9, -9, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
55. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 1], u=[1, 2], v=[1, 1], alpha=2, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(2, 1, [3, 5], -26, -26, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1, [3, 5], -26, -26, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
56. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-2, 2, [2, 6], 8, 8, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [2, 6], 8, 8, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
57. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, 3], u=[1, -1], v=[3, 1], alpha=-1, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-1, 3, [8, 4], 132, 132, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, [8, 4], 132, 132, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
58. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -3], u=[1, 0], v=[-3, 1], alpha=0, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(0, -3, [9, -3], 108, 108, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -3, [9, -3], 108, 108, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
59. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, -2], u=[1, 1], v=[-2, 1], alpha=1, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(1, -2, [5, -1], -30, -30, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [5, -1], -30, -30, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
60. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, -1], u=[1, 2], v=[-1, 1], alpha=2, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(2, -1, [3, 3], 0, 0, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -1, [3, 3], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
61. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 0], u=[1, -2], v=[0, 1], alpha=-2, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-2, 0, [-2, 4], -8, -8, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 0, [-2, 4], -8, -8, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
62. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, 1], u=[1, -1], v=[1, 1], alpha=-1, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-1, 1, [0, 2], -6, -6, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 1, [0, 2], -6, -6, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
63. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, 2], u=[1, 0], v=[2, 1], alpha=0, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(0, 2, [4, 2], -4, -4, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, [4, 2], -4, -4, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
64. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 3], u=[1, 1], v=[3, 1], alpha=1, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(1, 3, [10, 4], 172, 172, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, [10, 4], 172, 172, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
65. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, -3], u=[1, 2], v=[-3, 1], alpha=2, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(2, -3, [11, 1], 68, 68, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -3, [11, 1], 68, 68, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
66. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, -2], u=[1, -2], v=[-2, 1], alpha=-2, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-2, -2, [2, 2], -28, -28, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -2, [2, 2], -28, -28, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
67. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -1], u=[1, -1], v=[-1, 1], alpha=-1, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-1, -1, [0, 0], 0, 0, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -1, [0, 0], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
68. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, 0], u=[1, 0], v=[0, 1], alpha=0, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(0, 0, [0, 0], 0, 0, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0, 0], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
69. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 1], u=[1, 1], v=[1, 1], alpha=1, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(1, 1, [2, 2], 0, 0, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, [2, 2], 0, 0, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
70. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 2], u=[1, 2], v=[2, 1], alpha=2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(2, 2, [6, 6], 96, 96, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, [6, 6], 96, 96, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
71. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 3], u=[1, -2], v=[3, 1], alpha=-2, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-2, 3, [7, 7], -280, -280, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, [7, 7], -280, -280, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
72. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -3], u=[1, -1], v=[-3, 1], alpha=-1, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(-1, -3, [8, -2], -108, -108, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, [8, -2], -108, -108, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
73. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -2], u=[1, 0], v=[-2, 1], alpha=0, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(0, -2, [4, -2], 40, 40, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, [4, -2], 40, 40, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
74. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, -1], u=[1, 1], v=[-1, 1], alpha=1, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(1, -1, [2, 0], 22, 22, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, [2, 0], 22, 22, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
75. Linearity in the Second Slot with the First Slot Fixed
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 0], u=[1, 2], v=[0, 1], alpha=2, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).
View answer and explanation
(2, 0, [2, 4], 48, 48, Yes)
Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [2, 4], 48, 48, Yes).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-second-slot-identity · candidate
76. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-12, -3], [-3, -4]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3], [-3, -4]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
77. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-11, -2], [0, -3]], [0, 1, -2, 0], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2], [0, -3]], [0, 1, -2, 0], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
78. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-10, -1], [5, -2]], [0, 1, -1, 5], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1], [5, -2]], [0, 1, -1, 5], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
79. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-9, 0], [-1, -1]], [0, 1, 0, -1], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0], [-1, -1]], [0, 1, 0, -1], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
80. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [1, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-8, 1], [1, 0]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1], [1, 0]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
81. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-7, 2], [-2, 1]], [0, 1, 2, -2], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2], [-2, 1]], [0, 1, 2, -2], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
82. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-6, 3], [3, 2]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3], [3, 2]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
83. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-5, -3], [-3, 3]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3], [-3, 3]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
84. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [-2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-4, -2], [-2, 4]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2], [-2, 4]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
85. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-3, -1], [-4, -4]], [0, 1, -1, -4], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1], [-4, -4]], [0, 1, -1, -4], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
86. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-2, 0], [1, -3]], [0, 1, 0, 1], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0], [1, -3]], [0, 1, 0, 1], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
87. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[-1, 1], [-5, -2]], [0, 1, 1, -5], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1], [-5, -2]], [0, 1, 1, -5], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
88. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [2, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[0, 2], [2, -1]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2], [2, -1]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
89. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[1, 3], [5, 0]], [0, 1, 3, 5], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3], [5, 0]], [0, 1, 3, 5], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
90. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[2, -3], [-1, 1]], [0, 1, -3, -1], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3], [-1, 1]], [0, 1, -3, -1], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
91. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
92. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-1, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[4, -1], [-1, 3]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [-1, 3]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
93. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[5, 0], [3, 4]], [0, 1, 0, 3], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0], [3, 4]], [0, 1, 0, 3], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
94. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[6, 1], [-3, -4]], [0, 1, 1, -3], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1], [-3, -4]], [0, 1, 1, -3], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
95. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[7, 2], [2, -3]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2], [2, -3]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
96. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [3, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[8, 3], [3, -2]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3], [3, -2]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
97. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[9, -3], [1, -1]], [0, 1, -3, 1], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3], [1, -1]], [0, 1, -3, 1], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
98. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[10, -2], [-5, 0]], [0, 1, -2, -5], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2], [-5, 0]], [0, 1, -2, -5], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
99. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[11, -1], [0, 1]], [0, 1, -1, 0], No, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [0, 1]], [0, 1, -1, 0], No, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
100. Symmetry Is Additional to Bilinearity
Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [0, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
View answer and explanation
([[12, 0], [0, 2]], null, Yes, symmetry_is_extra_not_bilinearity)
Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0], [0, 2]], null, Yes, symmetry_is_extra_not_bilinearity).
Reviewed step-by-step derivation
Real bilinear form · qf-mod27-bilinear-symmetry-audit · candidate
Canonical question JSONL · English JSONL · Validation report