第 17071 題：線性映射的像

原題與已知條件

同一精確2×2線性映射與column-span證書。操作=image-membership；資料={"matrix":[[1,21],[2,42]],"target":[22,44]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。

• G001 — 同一精確2×2線性映射與column-span證書。操作=image-membership；資料={"matrix":[[1,21],[2,42]],"target":[22,44]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。

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-map-image

逐步推導

real map determinant diagonal product := mul(1, 42)

real map determinant diagonal product = 42

real map determinant off-diagonal product := mul(21, 2)

real map determinant off-diagonal product = 42

real map exact determinant := sub(42, 42)

real map exact determinant = 0

same ordered integer matrix=[[1, 21], [2, 42]];usual real map T:R2 to R2;original columns=[[1, 2], [21, 42]]

exact determinant=0;rank=1

primitive kernel first coordinate := div(21, 1)

primitive kernel first coordinate = 21

primitive kernel second coordinate := div(-1, 1)

primitive kernel second coordinate = -1

primitive kernel canonical sign coordinate0 := mul(1, 21)

primitive kernel canonical sign coordinate0 = 21

primitive kernel canonical sign coordinate1 := mul(1, -1)

primitive kernel canonical sign coordinate1 = -1

other row dependence factor := div(2, 1)

other row dependence factor = 2

other row dependence coordinate0 := mul(2, 1)

other row dependence coordinate0 = 2

other row dependence coordinate1 := mul(2, 21)

other row dependence coordinate1 = 42

FIRST nonzero original row index=0;row=[1, 21];gcd=1;raw integer kernel direction=[21, -1];canonical sign=1

primitive kernel basis=[[21, -1]];other original row=[2, 42]=(2)*selected row

canonical kernel basis vector row0 first term := mul(1, 21)

canonical kernel basis vector row0 first term = 21

canonical kernel basis vector row0 second term := mul(21, -1)

canonical kernel basis vector row0 second term = -21

canonical kernel basis vector image coordinate0 := sum(21, -21)

canonical kernel basis vector image coordinate0 = 0

canonical kernel basis vector row1 first term := mul(2, 21)

canonical kernel basis vector row1 first term = 42

canonical kernel basis vector row1 second term := mul(42, -1)

canonical kernel basis vector row1 second term = -42

canonical kernel basis vector image coordinate1 := sum(42, -42)

canonical kernel basis vector image coordinate1 = 0

one nonzero row equation(1)*x+(21)*y=0;nonzero verified kernel vector=[21, -1];chosen nonzero vector coordinate=0

EVERY real solution is lambda*[21, -1];choose lambda=solution_coordinate0/(21);conversely every real lambda multiple maps to0

other column dependence factor := div(21, 1)

other column dependence factor = 21

other column dependence coordinate0 := mul(21, 1)

other column dependence coordinate0 = 21

other column dependence coordinate1 := mul(21, 2)

other column dependence coordinate1 = 42

FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[21, 42]=(21)*basis

for ANY real(s,t),M(s,t)=((1)*s+(21)*t)*basis;EVERY lambda*basis obtained from input lambda*e0

real kernel source dimension minus rank := sub(2, 1)

real kernel source dimension minus rank = 1

same real map rank=1;full kernel basis=[[21, -1]];full image basis=[[1, 2]]

nullity=1;image basis cardinality=1;rank plus nullity=2

image rank1 cross first product := mul(1, 44)

image rank1 cross first product = 44

image rank1 cross second product := mul(2, 22)

image rank1 cross second product = 44

image rank1 column target cross := sub(44, 44)

image rank1 column target cross = 0

image rank1 target real coefficient := div(22, 1)

image rank1 target real coefficient = 22

image rank1 target representation coordinate0 := mul(22, 1)

image rank1 target representation coordinate0 = 22

image rank1 target representation coordinate1 := mul(22, 2)

image rank1 target representation coordinate1 = 44

original target=[22, 44];first nonzero original column=[1, 2];cross=0;real coefficient=22

target=coefficient*basis=[22, 44];real input coefficient*e0=[22, 0] gives target

逐步反向核對

• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 以精確有理數重算 image rank1 target representation coordinate1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 image rank1 target representation coordinate0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 image rank1 target real coefficient，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 image rank1 column target cross，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 image rank1 cross second product，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 image rank1 cross first product，不以小數近似替代。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 real kernel source dimension minus rank，不以小數近似替代。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 以精確有理數重算 other column dependence coordinate1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 other column dependence coordinate0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 other column dependence factor，不以小數近似替代。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector image coordinate1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector row1 second term，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector row1 first term，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector image coordinate0，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector row0 second term，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 canonical kernel basis vector row0 first term，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 other row dependence coordinate1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 other row dependence coordinate0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 other row dependence factor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 primitive kernel canonical sign coordinate1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 primitive kernel canonical sign coordinate0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 primitive kernel second coordinate，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 primitive kernel first coordinate，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 real map exact determinant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 real map determinant off-diagonal product，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 real map determinant diagonal product，不以小數近似替代。

結論

([22, 44], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 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。