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3B p10

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Signed-off-by: szdytom <szdytom@qq.com>
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方而静 2025-07-29 22:04:22 +08:00
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@ -160,3 +160,17 @@
#tab 这说明向量组 $T v_1, dots, T v_n$ 在 $W$ 中线性无关。
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#exercise_sol(type: "proof")[
设向量组 $v_1, dots, v_n$ 张成 $V$，$T in LinearMap(V, W)$，证明：$T v_1, dots, T v_n$ 张成 $range T$。
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设 $w in range T$，则可设 $v in V$，使得 $T v = w$。将 $v$ 表示为
$ v = z_1 v_1 + dots.c + z_n v_n $
#tab 其中 $z_1, dots, z_n in FF$。则
$ w = T v = T (z_1 v_1 + dots.c + z_n v_n) = z_1 T v_1 + dots.c + z_n T v_n $
#tab 这说明 $w$ 可以表示为向量组 $T v_1, dots, T v_n$ 的线性组合，根据张成的定义（原书2.7），可得 $T v_1, dots, T v_n$ 张成 $range T$。
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