Question 29674: One Exact Instance of Distributivity over Scalar Addition

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. a=-1, b=1, vector=[11, 2, -21]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

• 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. a=-1, b=1, vector=[11, 2, -21]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

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-distributivity, vectorspace-scalar-distributivity]

derived = [[a, b, vector, left, right, equal]]

Vector ORIGINAL scalar a plus b := sum(-1, 1)

Vector ORIGINAL scalar a plus b = 0

Vector ORIGINAL left distributed coordinate0 := mul(0, 11)

Vector ORIGINAL left distributed coordinate0 = 0

Vector ORIGINAL a term coordinate0 := mul(-1, 11)

Vector ORIGINAL a term coordinate0 = -11

Vector ORIGINAL b term coordinate0 := mul(1, 11)

Vector ORIGINAL b term coordinate0 = 11

Vector ORIGINAL right distributed coordinate0 := sum(-11, 11)

Vector ORIGINAL right distributed coordinate0 = 0

Vector ORIGINAL left distributed coordinate1 := mul(0, 2)

Vector ORIGINAL left distributed coordinate1 = 0

Vector ORIGINAL a term coordinate1 := mul(-1, 2)

Vector ORIGINAL a term coordinate1 = -2

Vector ORIGINAL b term coordinate1 := mul(1, 2)

Vector ORIGINAL b term coordinate1 = 2

Vector ORIGINAL right distributed coordinate1 := sum(-2, 2)

Vector ORIGINAL right distributed coordinate1 = 0

Vector ORIGINAL left distributed coordinate2 := mul(0, -21)

Vector ORIGINAL left distributed coordinate2 = 0

Vector ORIGINAL a term coordinate2 := mul(-1, -21)

Vector ORIGINAL a term coordinate2 = 21

Vector ORIGINAL b term coordinate2 := mul(1, -21)

Vector ORIGINAL b term coordinate2 = -21

Vector ORIGINAL right distributed coordinate2 := sum(21, -21)

Vector ORIGINAL right distributed coordinate2 = 0

source roles = [-1, 1, [11, 2, -21]]

derived = [[0, 0, 0], [0, 0, 0], Yes]

Reverse checks

• S015 · 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.
• S014 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate2 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Vector ORIGINAL scalar a plus b 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, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes)

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.