第 32805 題：矩陣模型加法兩側值

原題與已知條件

實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。u=[4, 1]、v=[-4, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。

• G001 — 實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。u=[4, 1]、v=[-4, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-linear-transformation

逐步推導

source roles = [qf-linear-map-additivity-values, additivity_values, reals, []]

derived = [[VECTOR_EQUALS_VECTOR_STRING]]

Additivity ORIGINAL u plus v ORIGINAL component0 := sum(4, -4)

Additivity ORIGINAL u plus v ORIGINAL component0 = 0

Additivity ORIGINAL u plus v ORIGINAL component1 := sum(1, 2)

Additivity ORIGINAL u plus v ORIGINAL component1 = 3

Additivity LEFT M of sum ORIGINAL row0 column0 product := mul(-8, 0)

Additivity LEFT M of sum ORIGINAL row0 column0 product = 0

Additivity LEFT M of sum ORIGINAL row0 column1 product := mul(1, 3)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 3

Additivity LEFT M of sum ORIGINAL row0 two products sum := sum(0, 3)

Additivity LEFT M of sum ORIGINAL row0 two products sum = 3

source roles = [Additivity LEFT M of sum, 0, [-8, 1], [0, 3]]

derived = [[0, 3], 3]

Additivity LEFT M of sum ORIGINAL row1 column0 product := mul(-2, 0)

Additivity LEFT M of sum ORIGINAL row1 column0 product = 0

Additivity LEFT M of sum ORIGINAL row1 column1 product := mul(5, 3)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 15

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 15)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 15

source roles = [Additivity LEFT M of sum, 1, [-2, 5], [0, 3]]

derived = [[0, 15], 15]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(-8, 4)

Additivity RIGHT M of u ORIGINAL row0 column0 product = -32

Additivity RIGHT M of u ORIGINAL row0 column1 product := mul(1, 1)

Additivity RIGHT M of u ORIGINAL row0 column1 product = 1

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(-32, 1)

Additivity RIGHT M of u ORIGINAL row0 two products sum = -31

source roles = [Additivity RIGHT M of u, 0, [-8, 1], [4, 1]]

derived = [[-32, 1], -31]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 4)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -8

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(5, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 5

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-8, 5)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -3

source roles = [Additivity RIGHT M of u, 1, [-2, 5], [4, 1]]

derived = [[-8, 5], -3]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(-8, -4)

Additivity RIGHT M of v ORIGINAL row0 column0 product = 32

Additivity RIGHT M of v ORIGINAL row0 column1 product := mul(1, 2)

Additivity RIGHT M of v ORIGINAL row0 column1 product = 2

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(32, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = 34

source roles = [Additivity RIGHT M of v, 0, [-8, 1], [-4, 2]]

derived = [[32, 2], 34]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -4)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 8

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(5, 2)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 10

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(8, 10)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 18

source roles = [Additivity RIGHT M of v, 1, [-2, 5], [-4, 2]]

derived = [[8, 10], 18]

Additivity RIGHT images sum ORIGINAL component0 := sum(-31, 34)

Additivity RIGHT images sum ORIGINAL component0 = 3

Additivity RIGHT images sum ORIGINAL component1 := sum(-3, 18)

Additivity RIGHT images sum ORIGINAL component1 = 15

source roles = [additivity_values, [[-8, 1], [-2, 5]], [4, 1], [-4, 2]]

derived = [[3, 15], [3, 15], [3, 15] = [3, 15]]

逐步反向核對

• S030 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S029 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT images sum ORIGINAL component1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT images sum ORIGINAL component0，不以小數近似替代。
• S027 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S026 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row1 two products sum，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row1 column1 product，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row1 column0 product，不以小數近似替代。
• S023 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S022 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row0 two products sum，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row0 column1 product，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of v ORIGINAL row0 column0 product，不以小數近似替代。
• S019 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S018 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row1 two products sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row1 column1 product，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row1 column0 product，不以小數近似替代。
• S015 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S014 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row0 two products sum，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row0 column1 product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Additivity RIGHT M of u ORIGINAL row0 column0 product，不以小數近似替代。
• S011 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S010 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row1 two products sum，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row1 column1 product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row1 column0 product，不以小數近似替代。
• S007 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S006 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row0 two products sum，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row0 column1 product，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Additivity LEFT M of sum ORIGINAL row0 column0 product，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Additivity ORIGINAL u plus v ORIGINAL component1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Additivity ORIGINAL u plus v ORIGINAL component0，不以小數近似替代。
• S001 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。

結論

[3, 15] = [3, 15]

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。