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3B p2

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Signed-off-by: szdytom <szdytom@qq.com>
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方而静 2025-07-29 18:38:01 +08:00
parent 6dbaebcb78
commit 572b6018b2
Signed by: szTom
GPG Key ID: 072D999D60C6473C
1 changed files with 11 additions and 1 deletions
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sections/3B.typ
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@ -1,5 +1,5 @@
#import "../styles.typ": exercise_sol, tab
#import "../math.typ": null, range
#import "../math.typ": null, range, LinearMap
#exercise_sol(type: "answer")[
给出一例：满足 $dim null T = 3$ 且 $dim range T = 2$ 的线性映射 $T$。
@ -15,3 +15,13 @@
#tab 于是 $dim null T = 3$ 且 $dim range T = 2$。
]
#exercise_sol(type: "proof")[
设 $S, T in LinearMap(V)$ 使得 $range S subset.eq null T$，证明：$(S T)^2 = 0$。
][
设 $v in V$。考虑到 $S (T v) in range S subset.eq null T$，根据定义，$ T S T v = 0$。根据线性映射将 $0$ 映射到 $0$（原书3.10），
$ (S T)^2 v = S (T S T) v = S 0 = 0 $
因此 $(S T)^2 = 0$。
]