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1C p19

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Signed-off-by: szdytom <szdytom@qq.com>
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方而静 2025-07-09 16:30:18 +08:00
parent 6ae61d52f4
commit c1cf2835ef
Signed by: szTom
GPG Key ID: 072D999D60C6473C
1 changed files with 20 additions and 1 deletions
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sections/1C.typ
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@ -399,7 +399,7 @@
] ]
#exercise_sol(type: "explain")[ #exercise_sol(type: "explain")[
设 令
$ U = {(x, -x, 2x) in FF^3 : x in FF} quad "与" quad W = {(x, x, 2x) in FF^3 : x in FF} $ $ U = {(x, -x, 2x) in FF^3 : x in FF} quad "与" quad W = {(x, x, 2x) in FF^3 : x in FF} $
@ -462,3 +462,22 @@
#tab 综上所述，“子空间求和”运算的加法单位元是 ${0}$，而只有 ${0}$ 有加法逆元。 #tab 综上所述，“子空间求和”运算的加法单位元是 ${0}$，而只有 ${0}$ 有加法逆元。
] ]
#exercise_sol(type: "answer")[
证明或证伪：如果 $V_1$，$V_2$，$U$ 都是 $V$ 的子空间，且
$ V_1 + U = V_2 + U $
则有 $V_1 = V_2$。
][
令
$ V &= RR^3 \
U &= {(0, x, y) in RR^3 : x,y in RR} \
V_1 &= {0} \
V_2 &= {(0, x, 0) in RR^3 : x in RR} $
#tab 注意到 $V_1 + U = U = V_2 + U$，但 $V_1 != V_2$。
#tab 该反例说明，题目中的命题不成立。
]