第 32812 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=[[-1, 4], [-2, 12]]，T(x)=Mx。u=[11, 1]、v=[-11, -1]，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=[[-1, 4], [-2, 12]]，T(x)=Mx。u=[11, 1]、v=[-11, -1]，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(11, -11)

Additivity ORIGINAL u plus v ORIGINAL component0 = 0

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

Additivity ORIGINAL u plus v ORIGINAL component1 = 0

Additivity LEFT M of sum ORIGINAL row0 column0 product := mul(-1, 0)

Additivity LEFT M of sum ORIGINAL row0 column0 product = 0

Additivity LEFT M of sum ORIGINAL row0 column1 product := mul(4, 0)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 0

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

Additivity LEFT M of sum ORIGINAL row0 two products sum = 0

source roles = [Additivity LEFT M of sum, 0, [-1, 4], [0, 0]]

derived = [[0, 0], 0]

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(12, 0)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 0

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

Additivity LEFT M of sum ORIGINAL row1 two products sum = 0

source roles = [Additivity LEFT M of sum, 1, [-2, 12], [0, 0]]

derived = [[0, 0], 0]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(-1, 11)

Additivity RIGHT M of u ORIGINAL row0 column0 product = -11

Additivity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 1)

Additivity RIGHT M of u ORIGINAL row0 column1 product = 4

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(-11, 4)

Additivity RIGHT M of u ORIGINAL row0 two products sum = -7

source roles = [Additivity RIGHT M of u, 0, [-1, 4], [11, 1]]

derived = [[-11, 4], -7]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 11)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -22

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(12, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 12

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-22, 12)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -10

source roles = [Additivity RIGHT M of u, 1, [-2, 12], [11, 1]]

derived = [[-22, 12], -10]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(-1, -11)

Additivity RIGHT M of v ORIGINAL row0 column0 product = 11

Additivity RIGHT M of v ORIGINAL row0 column1 product := mul(4, -1)

Additivity RIGHT M of v ORIGINAL row0 column1 product = -4

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(11, -4)

Additivity RIGHT M of v ORIGINAL row0 two products sum = 7

source roles = [Additivity RIGHT M of v, 0, [-1, 4], [-11, -1]]

derived = [[11, -4], 7]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -11)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 22

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(12, -1)

Additivity RIGHT M of v ORIGINAL row1 column1 product = -12

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(22, -12)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 10

source roles = [Additivity RIGHT M of v, 1, [-2, 12], [-11, -1]]

derived = [[22, -12], 10]

Additivity RIGHT images sum ORIGINAL component0 := sum(-7, 7)

Additivity RIGHT images sum ORIGINAL component0 = 0

Additivity RIGHT images sum ORIGINAL component1 := sum(-10, 10)

Additivity RIGHT images sum ORIGINAL component1 = 0

source roles = [additivity_values, [[-1, 4], [-2, 12]], [11, 1], [-11, -1]]

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

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

[0, 0] = [0, 0]

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。