szTom/LADRSolutions
1
0
Fork 0
mirror of https://github.com/szdytom/LADRSolutions.git synced 2025-10-19 16:30:16 +00:00
Code Issues Packages Projects Releases Wiki Activity

3B p4

Browse Source 
Signed-off-by: szdytom <szdytom@qq.com>
This commit is contained in:
方而静 2025-07-29 19:56:16 +08:00
parent 94ad1d39e6
commit 38bdc65d1a
Signed by: szTom
GPG Key ID: 072D999D60C6473C
1 changed files with 15 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

15
sections/3B.typ
View File

@ -42,3 +42,18 @@
#tab 现在假设向量组 $v_1, dots, v_m$ 线性无关。则 $dim range T = dim span(v_1, dots, v_m) = m$，根据线性映射基本定理（原书3.21），$dim FF^m = dim null T + dim range T$，解得 $dim null T = {0}$，即 $T$ 是单射。
]
#exercise_sol(type: "proof")[
证明：${T in LinearMap(RR^5, RR^4) : dim null T > 2}$ 不是 $LinearMap(RR^5, RR^4)$ 的子空间。
][
记 $S = {T in LinearMap(RR^5, RR^4) : dim null T > 2}$。取 $T_1, T_2 in LinearMap(RR^5, RR^4)$，使得对于任意 $x_1, x_2, x_3, x_4, x_5 in RR$，有
$ T_1(x_1, x_2, x_3, x_4, x_5) &= (x_1, x_2, 0, 0) \
T_2(x_1, x_2, x_3, x_4, x_5) &= (0, 0, x_3, x_4) $
容易验证 $dim null T_1 = dim null T_2 = 3 > 2$，即 $T_1$ 和 $T_2$ 都是 $S$ 中的元素。然而，注意到 $dim range (T_1 + T_2) = 4$，即根据线性映射基本定理（原书3.21），
$ dim null (T_1 + T_2) = dim RR^5 - dim range (T_1 + T_2) = 5 - 4 = 1 $
#tab 因此 $T_1 + T_2 in.not S$。这说明 $S$ 违反了子空间的条件（原书1.34）中对加法封闭性的要求，故 $S$ 不是 $LinearMap(RR^5, RR^4)$ 的子空间。
]