Question 15235: Compute Distance in the Norm Induced by This Positive-Definite Form

Original question and givens

Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-3, 2], right vector=[3, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

• G001 — Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-3, 2], right vector=[3, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-real-hilbert-space

Step-by-step derivation

Hilbert Gram diagonal product := mul(11, 1)

Hilbert Gram diagonal product = 11

Hilbert Gram off-diagonal product := mul(0, 0)

Hilbert Gram off-diagonal product = 0

Hilbert Gram determinant := sub(11, 0)

Hilbert Gram determinant = 11

Hilbert completing-square slope := div(0, 11)

Hilbert completing-square slope = 0

Hilbert completing-square positive residual := div(11, 11)

Hilbert completing-square positive residual = 1

complete real carrier R^2;Gram matrix=[[11, 0], [0, 1]];a=11>0;det=11>0;dimension declaration=2

x^T M x=11*(x0+(0)*x1)^2+(1)*x1^2;zero only at x=0

Hilbert distance coordinate0 difference := sub(-3, 3)

Hilbert distance coordinate0 difference = -6

Hilbert distance coordinate1 difference := sub(2, -1)

Hilbert distance coordinate1 difference = 3

Hilbert distance coefficient(0,0) times left coordinate := mul(11, -6)

Hilbert distance coefficient(0,0) times left coordinate = -66

Hilbert distance term(0,0) := mul(-66, -6)

Hilbert distance term(0,0) = 396

Hilbert distance coefficient(0,1) times left coordinate := mul(0, -6)

Hilbert distance coefficient(0,1) times left coordinate = 0

Hilbert distance term(0,1) := mul(0, 3)

Hilbert distance term(0,1) = 0

Hilbert distance coefficient(1,0) times left coordinate := mul(0, 3)

Hilbert distance coefficient(1,0) times left coordinate = 0

Hilbert distance term(1,0) := mul(0, -6)

Hilbert distance term(1,0) = 0

Hilbert distance coefficient(1,1) times left coordinate := mul(1, 3)

Hilbert distance coefficient(1,1) times left coordinate = 3

Hilbert distance term(1,1) := mul(3, 3)

Hilbert distance term(1,1) = 9

Hilbert distance complete bilinear sum := sum(396, 0, 0, 9)

Hilbert distance complete bilinear sum = 405

left=[-3, 2];right=[3, -1];same induced metric on R^2;squared norm=405

distance=sqrt(405)

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Hilbert distance complete bilinear sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Hilbert distance term(1,1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Hilbert distance term(1,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Hilbert distance term(0,1) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Hilbert distance term(0,0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Hilbert distance coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Hilbert distance coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals.

Result

([-3, 2], [3, -1], [-6, 3], 405, sqrt(405))

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.