Question 29694: A Declared Real-Scalar Instance on Integer Coordinates

Original question and givens

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[6, 1, -15]. Give (scalar,vector,scaled vector).

• G001 — Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[6, 1, -15]. Give (scalar,vector,scaled vector).

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-vector-space

Step-by-step derivation

source roles = [qf-str22-vectorspace-scalar-multiplication, vectorspace-scalar-multiplication]

derived = [[scalar, vector, result]]

Vector ORIGINAL scalar product coordinate0 := mul(1, 6)

Vector ORIGINAL scalar product coordinate0 = 6

Vector ORIGINAL scalar product coordinate1 := mul(1, 1)

Vector ORIGINAL scalar product coordinate1 = 1

Vector ORIGINAL scalar product coordinate2 := mul(1, -15)

Vector ORIGINAL scalar product coordinate2 = -15

source roles = [1, [6, 1, -15]]

derived = [[6, 1, -15]]

Reverse checks

• S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S004 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.

Result

(1, [6, 1, -15], [6, 1, -15])

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.