Original question and givens
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).
G001 — 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).
Step-by-step derivation
S001 · implies
Before
fixed real matrix=[[-11, -2], [0, -3]];B(x,y)=sum_i sum_j x_i*A_ij*y_j
After
distributivity proves real linearity in each slot separately for all vectors and scalars
Conditions: none
Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling.
Step identifiers and dependencies
Rule: matrix_formula_bilinearity_universal_not_samples · Uses: G001
S002 · implies
Before
requested form:matrix=[[-11, -2], [0, -3]];first slot=[-1, -2];second slot=[-4, 11]
After
B(first,second)=first^T*(matrix*second);slot order fixed
Conditions: none
First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged.
Step identifiers and dependencies
Rule: bilinear_ordered_matrix_slots · Uses: G001, S001
S003 · calculation
Before
requested form matrix row0 column0 term := mul(-11, -4)
After
requested form matrix row0 column0 term = 44
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
requested form matrix row0 column1 term := mul(-2, 11)
After
requested form matrix row0 column1 term = -22
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
requested form matrix-times-second row0 := sum(44, -22)
After
requested form matrix-times-second row0 = 22
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
requested form matrix row1 column0 term := mul(0, -4)
After
requested form matrix row1 column0 term = 0
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
requested form matrix row1 column1 term := mul(-3, 11)
After
requested form matrix row1 column1 term = -33
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
requested form matrix-times-second row1 := sum(0, -33)
After
requested form matrix-times-second row1 = -33
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
requested form first-slot coordinate0 term := mul(-1, 22)
After
requested form first-slot coordinate0 term = -22
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
requested form first-slot coordinate1 term := mul(-2, -33)
After
requested form first-slot coordinate1 term = 66
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
requested form final first-slot dot sum := sum(-22, 66)
After
requested form final first-slot dot sum = 44
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
Result
([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.