第 19982 題：商向量空間

原題與已知條件

有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={"ambient_dimension":3,"basis_w":[[1,7,0]],"v":[8,2,3],"u":[3,7,4],"shift_v_coefficients":[8],"shift_u_coefficients":[-7],"scalar":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={"ambient_dimension":3,"basis_w":[[1,7,0]],"v":[8,2,3],"u":[3,7,4],"shift_v_coefficients":[8],"shift_u_coefficients":[-7],"scalar":7}。請依 exact_ordered_tuple_declared_by_interface 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-quotient-vector-space

逐步推導

source operation=quotient-representative-independence;original snapshot and mathematics retained

explicit ordered source output fields=[shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate]

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [7], [0]];rows=3;columns=1

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1)

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1

quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1]

normalized FULL row=[1]

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(7, 1)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 7

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(7, 7)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0

quotient ORIGINAL ordered W basis:target row=1;old row=[7];subtract factor=7 times FULL pivot row=[1]

new exact FULL row=[0]

quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 7, 0]]

W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone

ACTUAL subspace W full real span in R^3;original basis=[[1, 7, 0]]

for ANY representatives v'=v+w,u'=u+z with w,z in W,and ANY real lambda,sum difference=w+z in W and scalar difference=lambda*w in W;thus quotient addition/scaling independent of representatives

quotient original v shift original coordinate0 coefficient0 product := mul(1, 8)

quotient original v shift original coordinate0 coefficient0 product = 8

quotient original v shift original coordinate0 complete sum := sum(8)

quotient original v shift original coordinate0 complete sum = 8

quotient original v shift original coordinate1 coefficient0 product := mul(7, 8)

quotient original v shift original coordinate1 coefficient0 product = 56

quotient original v shift original coordinate1 complete sum := sum(56)

quotient original v shift original coordinate1 complete sum = 56

quotient original v shift original coordinate2 coefficient0 product := mul(0, 8)

quotient original v shift original coordinate2 coefficient0 product = 0

quotient original v shift original coordinate2 complete sum := sum(0)

quotient original v shift original coordinate2 complete sum = 0

quotient original v shift:ORIGINAL W basis order=[[1, 7, 0]];aligned coefficient block=[8]

computed whole vector=[8, 56, 0];a real linear combination in W by its declared span

quotient original u shift original coordinate0 coefficient0 product := mul(1, -7)

quotient original u shift original coordinate0 coefficient0 product = -7

quotient original u shift original coordinate0 complete sum := sum(-7)

quotient original u shift original coordinate0 complete sum = -7

quotient original u shift original coordinate1 coefficient0 product := mul(7, -7)

quotient original u shift original coordinate1 coefficient0 product = -49

quotient original u shift original coordinate1 complete sum := sum(-49)

quotient original u shift original coordinate1 complete sum = -49

quotient original u shift original coordinate2 coefficient0 product := mul(0, -7)

quotient original u shift original coordinate2 coefficient0 product = 0

quotient original u shift original coordinate2 complete sum := sum(0)

quotient original u shift original coordinate2 complete sum = 0

quotient original u shift:ORIGINAL W basis order=[[1, 7, 0]];aligned coefficient block=[-7]

computed whole vector=[-7, -49, 0];a real linear combination in W by its declared span

quotient changed v coordinate0 := sum(8, 8)

quotient changed v coordinate0 = 16

quotient changed v coordinate1 := sum(2, 56)

quotient changed v coordinate1 = 58

quotient changed v coordinate2 := sum(3, 0)

quotient changed v coordinate2 = 3

quotient changed u coordinate0 := sum(3, -7)

quotient changed u coordinate0 = -4

quotient changed u coordinate1 := sum(7, -49)

quotient changed u coordinate1 = -42

quotient changed u coordinate2 := sum(4, 0)

quotient changed u coordinate2 = 4

quotient new sum coordinate0 := sum(16, -4)

quotient new sum coordinate0 = 12

quotient old sum coordinate0 := sum(8, 3)

quotient old sum coordinate0 = 11

quotient changed sum difference coordinate0 := sub(12, 11)

quotient changed sum difference coordinate0 = 1

quotient new scalar representative coordinate0 := mul(7, 16)

quotient new scalar representative coordinate0 = 112

quotient old scalar representative coordinate0 := mul(7, 8)

quotient old scalar representative coordinate0 = 56

quotient changed scalar difference coordinate0 := sub(112, 56)

quotient changed scalar difference coordinate0 = 56

quotient new sum coordinate1 := sum(58, -42)

quotient new sum coordinate1 = 16

quotient old sum coordinate1 := sum(2, 7)

quotient old sum coordinate1 = 9

quotient changed sum difference coordinate1 := sub(16, 9)

quotient changed sum difference coordinate1 = 7

quotient new scalar representative coordinate1 := mul(7, 58)

quotient new scalar representative coordinate1 = 406

quotient old scalar representative coordinate1 := mul(7, 2)

quotient old scalar representative coordinate1 = 14

quotient changed scalar difference coordinate1 := sub(406, 14)

quotient changed scalar difference coordinate1 = 392

quotient new sum coordinate2 := sum(3, 4)

quotient new sum coordinate2 = 7

quotient old sum coordinate2 := sum(3, 4)

quotient old sum coordinate2 = 7

quotient changed sum difference coordinate2 := sub(7, 7)

quotient changed sum difference coordinate2 = 0

quotient new scalar representative coordinate2 := mul(7, 3)

quotient new scalar representative coordinate2 = 21

quotient old scalar representative coordinate2 := mul(7, 3)

quotient old scalar representative coordinate2 = 21

quotient changed scalar difference coordinate2 := sub(21, 21)

quotient changed scalar difference coordinate2 = 0

quotient supplied v shift membership:COMPLETE exact original ordered matrix=[[1, 8], [7, 56], [0, 0]];rows=3;columns=2

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient supplied v shift membership pivot0 column0 normalized entry0 := div(1, 1)

quotient supplied v shift membership pivot0 column0 normalized entry0 = 1

quotient supplied v shift membership pivot0 column0 normalized entry1 := div(8, 1)

quotient supplied v shift membership pivot0 column0 normalized entry1 = 8

quotient supplied v shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 8]

normalized FULL row=[1, 8]

quotient supplied v shift membership eliminate row1 pivot0 entry0 product := mul(7, 1)

quotient supplied v shift membership eliminate row1 pivot0 entry0 product = 7

quotient supplied v shift membership eliminate row1 pivot0 entry0 difference := sub(7, 7)

quotient supplied v shift membership eliminate row1 pivot0 entry0 difference = 0

quotient supplied v shift membership eliminate row1 pivot0 entry1 product := mul(7, 8)

quotient supplied v shift membership eliminate row1 pivot0 entry1 product = 56

quotient supplied v shift membership eliminate row1 pivot0 entry1 difference := sub(56, 56)

quotient supplied v shift membership eliminate row1 pivot0 entry1 difference = 0

quotient supplied v shift membership:target row=1;old row=[7, 56];subtract factor=7 times FULL pivot row=[1, 8]

new exact FULL row=[0, 0]

quotient supplied v shift membership:column=1;remaining exact entries=[0, 0]

no pivot added;current rank=1

quotient supplied v shift membership:complete exact reduced matrix=[[1, 8], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product := mul(1, 8)

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product = 8

quotient supplied v shift membership original coordinate0 reconstruction complete sum := sum(8)

quotient supplied v shift membership original coordinate0 reconstruction complete sum = 8

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product := mul(7, 8)

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product = 56

quotient supplied v shift membership original coordinate1 reconstruction complete sum := sum(56)

quotient supplied v shift membership original coordinate1 reconstruction complete sum = 56

quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product := mul(0, 8)

quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product = 0

quotient supplied v shift membership original coordinate2 reconstruction complete sum := sum(0)

quotient supplied v shift membership original coordinate2 reconstruction complete sum = 0

quotient supplied v shift membership:original proposed vector=[8, 56, 0];original ordered independent columns=[[1, 7, 0]];complete reduced augmented system=[[1, 8], [0, 0], [0, 0]]

exact real-span coefficient witness=[8];FULL original reconstruction=[8, 56, 0];membership=是

quotient supplied u shift membership:COMPLETE exact original ordered matrix=[[1, -7], [7, -49], [0, 0]];rows=3;columns=2

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient supplied u shift membership pivot0 column0 normalized entry0 := div(1, 1)

quotient supplied u shift membership pivot0 column0 normalized entry0 = 1

quotient supplied u shift membership pivot0 column0 normalized entry1 := div(-7, 1)

quotient supplied u shift membership pivot0 column0 normalized entry1 = -7

quotient supplied u shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -7]

normalized FULL row=[1, -7]

quotient supplied u shift membership eliminate row1 pivot0 entry0 product := mul(7, 1)

quotient supplied u shift membership eliminate row1 pivot0 entry0 product = 7

quotient supplied u shift membership eliminate row1 pivot0 entry0 difference := sub(7, 7)

quotient supplied u shift membership eliminate row1 pivot0 entry0 difference = 0

quotient supplied u shift membership eliminate row1 pivot0 entry1 product := mul(7, -7)

quotient supplied u shift membership eliminate row1 pivot0 entry1 product = -49

quotient supplied u shift membership eliminate row1 pivot0 entry1 difference := sub(-49, -49)

quotient supplied u shift membership eliminate row1 pivot0 entry1 difference = 0

quotient supplied u shift membership:target row=1;old row=[7, -49];subtract factor=7 times FULL pivot row=[1, -7]

new exact FULL row=[0, 0]

quotient supplied u shift membership:column=1;remaining exact entries=[0, 0]

no pivot added;current rank=1

quotient supplied u shift membership:complete exact reduced matrix=[[1, -7], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product := mul(1, -7)

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product = -7

quotient supplied u shift membership original coordinate0 reconstruction complete sum := sum(-7)

quotient supplied u shift membership original coordinate0 reconstruction complete sum = -7

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product := mul(7, -7)

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product = -49

quotient supplied u shift membership original coordinate1 reconstruction complete sum := sum(-49)

quotient supplied u shift membership original coordinate1 reconstruction complete sum = -49

quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product := mul(0, -7)

quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product = 0

quotient supplied u shift membership original coordinate2 reconstruction complete sum := sum(0)

quotient supplied u shift membership original coordinate2 reconstruction complete sum = 0

quotient supplied u shift membership:original proposed vector=[-7, -49, 0];original ordered independent columns=[[1, 7, 0]];complete reduced augmented system=[[1, -7], [0, 0], [0, 0]]

exact real-span coefficient witness=[-7];FULL original reconstruction=[-7, -49, 0];membership=是

quotient supplied sum difference membership:COMPLETE exact original ordered matrix=[[1, 1], [7, 7], [0, 0]];rows=3;columns=2

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient supplied sum difference membership pivot0 column0 normalized entry0 := div(1, 1)

quotient supplied sum difference membership pivot0 column0 normalized entry0 = 1

quotient supplied sum difference membership pivot0 column0 normalized entry1 := div(1, 1)

quotient supplied sum difference membership pivot0 column0 normalized entry1 = 1

quotient supplied sum difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 1]

normalized FULL row=[1, 1]

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product := mul(7, 1)

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product = 7

quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference := sub(7, 7)

quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference = 0

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product := mul(7, 1)

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product = 7

quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference := sub(7, 7)

quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference = 0

quotient supplied sum difference membership:target row=1;old row=[7, 7];subtract factor=7 times FULL pivot row=[1, 1]

new exact FULL row=[0, 0]

quotient supplied sum difference membership:column=1;remaining exact entries=[0, 0]

no pivot added;current rank=1

quotient supplied sum difference membership:complete exact reduced matrix=[[1, 1], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 1)

quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product = 1

quotient supplied sum difference membership original coordinate0 reconstruction complete sum := sum(1)

quotient supplied sum difference membership original coordinate0 reconstruction complete sum = 1

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product := mul(7, 1)

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product = 7

quotient supplied sum difference membership original coordinate1 reconstruction complete sum := sum(7)

quotient supplied sum difference membership original coordinate1 reconstruction complete sum = 7

quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 1)

quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product = 0

quotient supplied sum difference membership original coordinate2 reconstruction complete sum := sum(0)

quotient supplied sum difference membership original coordinate2 reconstruction complete sum = 0

quotient supplied sum difference membership:original proposed vector=[1, 7, 0];original ordered independent columns=[[1, 7, 0]];complete reduced augmented system=[[1, 1], [0, 0], [0, 0]]

exact real-span coefficient witness=[1];FULL original reconstruction=[1, 7, 0];membership=是

quotient supplied scalar difference membership:COMPLETE exact original ordered matrix=[[1, 56], [7, 392], [0, 0]];rows=3;columns=2

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient supplied scalar difference membership pivot0 column0 normalized entry0 := div(1, 1)

quotient supplied scalar difference membership pivot0 column0 normalized entry0 = 1

quotient supplied scalar difference membership pivot0 column0 normalized entry1 := div(56, 1)

quotient supplied scalar difference membership pivot0 column0 normalized entry1 = 56

quotient supplied scalar difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 56]

normalized FULL row=[1, 56]

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product := mul(7, 1)

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product = 7

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference := sub(7, 7)

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference = 0

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product := mul(7, 56)

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product = 392

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference := sub(392, 392)

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference = 0

quotient supplied scalar difference membership:target row=1;old row=[7, 392];subtract factor=7 times FULL pivot row=[1, 56]

new exact FULL row=[0, 0]

quotient supplied scalar difference membership:column=1;remaining exact entries=[0, 0]

no pivot added;current rank=1

quotient supplied scalar difference membership:complete exact reduced matrix=[[1, 56], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 56)

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product = 56

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum := sum(56)

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum = 56

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product := mul(7, 56)

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product = 392

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum := sum(392)

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum = 392

quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 56)

quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product = 0

quotient supplied scalar difference membership original coordinate2 reconstruction complete sum := sum(0)

quotient supplied scalar difference membership original coordinate2 reconstruction complete sum = 0

quotient supplied scalar difference membership:original proposed vector=[56, 392, 0];original ordered independent columns=[[1, 7, 0]];complete reduced augmented system=[[1, 56], [0, 0], [0, 0]]

exact real-span coefficient witness=[56];FULL original reconstruction=[56, 392, 0];membership=是

ACTUAL original W shifts=[[8, 56, 0], [-7, -49, 0]];changed representatives=[[16, 58, 3], [-4, -42, 4]];actual sum/scalar differences=[[1, 7, 0], [56, 392, 0]]

ALL four exact full-span memberships=[是, 是, 是, 是];differences equal shift_v+shift_u and lambda*shift_v,matching the arbitrary-representative proof

逐步反向核對

• S121 · 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。
• S120 · 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。
• S119 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate2 reconstruction complete sum，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate1 reconstruction complete sum，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate0 reconstruction complete sum，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。
• S113 · 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。
• S112 · 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。
• S111 · 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。
• S110 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S106 · 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。
• S105 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership pivot0 column0 normalized entry1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 quotient supplied scalar difference membership pivot0 column0 normalized entry0，不以小數近似替代。
• S103 · 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。
• S102 · 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。
• S101 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate2 reconstruction complete sum，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate1 reconstruction complete sum，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate0 reconstruction complete sum，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership eliminate row1 pivot0 entry1 product，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S088 · 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。
• S087 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership pivot0 column0 normalized entry1，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 quotient supplied sum difference membership pivot0 column0 normalized entry0，不以小數近似替代。
• S085 · 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。
• S084 · 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。
• S083 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate2 reconstruction complete sum，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate1 reconstruction complete sum，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate0 reconstruction complete sum，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership eliminate row1 pivot0 entry1 product，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership pivot0 column0 normalized entry1，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 quotient supplied u shift membership pivot0 column0 normalized entry0，不以小數近似替代。
• S067 · 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。
• S066 · 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。
• S065 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate2 reconstruction complete sum，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate1 reconstruction complete sum，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate0 reconstruction complete sum，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership eliminate row1 pivot0 entry1 product，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S052 · 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。
• S051 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership pivot0 column0 normalized entry1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 quotient supplied v shift membership pivot0 column0 normalized entry0，不以小數近似替代。
• S049 · 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。
• S048 · premise_and_result_audit — 以精確有理數重算 quotient changed scalar difference coordinate2，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 quotient old scalar representative coordinate2，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 quotient new scalar representative coordinate2，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 quotient changed sum difference coordinate2，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 quotient old sum coordinate2，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 quotient new sum coordinate2，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 quotient changed scalar difference coordinate1，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 quotient old scalar representative coordinate1，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 quotient new scalar representative coordinate1，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 quotient changed sum difference coordinate1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 quotient old sum coordinate1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 quotient new sum coordinate1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 quotient changed scalar difference coordinate0，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 quotient old scalar representative coordinate0，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 quotient new scalar representative coordinate0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 quotient changed sum difference coordinate0，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 quotient old sum coordinate0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 quotient new sum coordinate0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 quotient changed u coordinate2，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 quotient changed u coordinate1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 quotient changed u coordinate0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 quotient changed v coordinate2，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 quotient changed v coordinate1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 quotient changed v coordinate0，不以小數近似替代。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate2 complete sum，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate2 coefficient0 product，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate1 complete sum，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate1 coefficient0 product，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate0 complete sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 quotient original u shift original coordinate0 coefficient0 product，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate2 complete sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate2 coefficient0 product，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate1 complete sum，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate1 coefficient0 product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate0 complete sum，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 quotient original v shift original coordinate0 coefficient0 product，不以小數近似替代。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product，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 quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0，不以小數近似替代。
• S002 · 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。
• S001 · 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。

結論

(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)

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。