第 32814 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, 2], [-2, 14]]，T(x)=Mx。u=[13, 1]、v=[-13, 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, 2], [-2, 14]]，T(x)=Mx。u=[13, 1]、v=[-13, 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(13, -13)

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 = 2

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(2, 2)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 4

Additivity LEFT M of sum ORIGINAL row0 two products sum := sum(0, 4)

Additivity LEFT M of sum ORIGINAL row0 two products sum = 4

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

derived = [[0, 4], 4]

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(14, 2)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 28

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 28)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 28

source roles = [Additivity LEFT M of sum, 1, [-2, 14], [0, 2]]

derived = [[0, 28], 28]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(1, 13)

Additivity RIGHT M of u ORIGINAL row0 column0 product = 13

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

Additivity RIGHT M of u ORIGINAL row0 column1 product = 2

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(13, 2)

Additivity RIGHT M of u ORIGINAL row0 two products sum = 15

source roles = [Additivity RIGHT M of u, 0, [1, 2], [13, 1]]

derived = [[13, 2], 15]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 13)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -26

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(14, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 14

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-26, 14)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -12

source roles = [Additivity RIGHT M of u, 1, [-2, 14], [13, 1]]

derived = [[-26, 14], -12]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(1, -13)

Additivity RIGHT M of v ORIGINAL row0 column0 product = -13

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

Additivity RIGHT M of v ORIGINAL row0 column1 product = 2

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(-13, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = -11

source roles = [Additivity RIGHT M of v, 0, [1, 2], [-13, 1]]

derived = [[-13, 2], -11]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -13)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 26

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(14, 1)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 14

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(26, 14)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 40

source roles = [Additivity RIGHT M of v, 1, [-2, 14], [-13, 1]]

derived = [[26, 14], 40]

Additivity RIGHT images sum ORIGINAL component0 := sum(15, -11)

Additivity RIGHT images sum ORIGINAL component0 = 4

Additivity RIGHT images sum ORIGINAL component1 := sum(-12, 40)

Additivity RIGHT images sum ORIGINAL component1 = 28

source roles = [additivity_values, [[1, 2], [-2, 14]], [13, 1], [-13, 1]]

derived = [[4, 28], [4, 28], [4, 28] = [4, 28]]

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

[4, 28] = [4, 28]

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。