From f3ee3fe44ca9aed8325a23178bb9699945d14be9 Mon Sep 17 00:00:00 2001 From: szdytom Date: Mon, 28 Jul 2025 23:44:11 +0800 Subject: [PATCH] 3A p15 Signed-off-by: szdytom --- sections/3A.typ | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/sections/3A.typ b/sections/3A.typ index 4c40770..bcf62e5 100644 --- a/sections/3A.typ +++ b/sections/3A.typ @@ -369,3 +369,17 @@ #tab 所以，根据#exercise_ref()，$LinearMap(V, W)$ 是无限维的。 ] + +#exercise_sol(type: "proof")[ + 设 $v_1, dots, v_m$ 是 $V$ 中的线性相关向量组，$dim W > 0$。证明：存在 $w_1, dots, w_m in W$，使得不存在 $T in LinearMap(V, W)$ 对于任意 $k in {1, dots, m}$，都有 $T v_k = w_k$。 +][ + 根据线性相关性引理（原书2.19），存在 $k in {1, dots, m}$，使得 $v_k in span(v_1, dots, v_(k - 1))$。设 + + $ v_k = a_1 v_1 + dots.c + a_(k - 1) v_(k - 1) $ + + #tab 其中 $a_1, dots, a_(k - 1) in FF$。任取 $w_k != 0$，并令 $w_1 = dots.c = w_(k - 1) = 0$。于是 + + $ T v_k = a_1 T v_1 + dots.c + a_(k - 1) T v_(k - 1) = 0 != w_k $ + + #tab 这说明不存在 $T in LinearMap(V, W)$，使得对于任意 $k in {1, dots, m}$，都有 $T v_k = w_k$。 +]