题目1

定义：

$\sigma \in \mathcal{L}(V_1,V_2)$（$\sigma$是线性空间$V_1$到$V_2$的一个线性映射）

$I_m(\sigma)\triangleq\{\sigma(\alpha)\mid\alpha\in V_1\}$（$I_m$是$\sigma$的像/值域）

$\operatorname{Ker}(\sigma)\triangleq\{\alpha\in V_1\mid\sigma(\alpha)=0\}$（$\operatorname{Ker}(\sigma)$是$\sigma$的核）

证明：

1. $I_m(\sigma)$是$V_2$的子空间，$\operatorname{Ker}(\sigma)$是$V_1$的子空间
2. $I_m(\sigma)=\operatorname{span}(\sigma(\alpha_1),\sigma(\alpha_2),\cdots,\sigma(\alpha_n))$，其中$\alpha_1,\alpha_2,\cdots,\alpha_n$是$V_1$的基
3. $\operatorname{dim}(I_m(\sigma))=\operatorname{rank}(A)$，其中$A$是$\sigma$的矩阵
4. $\operatorname{dim}(I_m(\sigma))+\operatorname{dim}(\operatorname{Ker}(\sigma))=n$

解：

(1) 证明子空间只需要验证加法与数乘封闭性

即$\forall \alpha,\beta \in V_1, k \in \mathbb{P}$

由线性映射的定义有，$\sigma(\alpha)+\sigma(\beta)=\sigma(\alpha+\beta)\in I_m(\sigma)$，$k\sigma(\alpha)=\sigma(k\alpha)\in I_m(\sigma)$，故$I_m(\sigma)$是$V_2$的子空间

即$\forall \alpha,\beta \in \operatorname{Ker}(\sigma), k \in \mathbb{P}$

有，$\sigma(\alpha)+\sigma(\beta)=\sigma(\alpha+\beta)=0$，$\sigma(k\alpha)=k\sigma(\alpha)=0$，故$\operatorname{Ker}(\sigma)$是$V_1$的子空间

(2)$\forall \alpha \in V_1$有，

则

也即$\forall \beta \in I_m(\sigma),\beta=\sum_{i=1}^{n}x_i\sigma(\alpha_i)$，故$I_m(\sigma)=\operatorname{span}(\sigma(\alpha_1),\sigma(\alpha_2),\cdots,\sigma(\alpha_n))$

(3)由于$\operatorname{dim}(I_m(\sigma))=\operatorname{rank}(\sigma(\alpha_1),\sigma(\alpha_2),\cdots,\sigma(\alpha_n))$ 而$\operatorname{rank}(\sigma(\alpha_1),\sigma(\alpha_2),\cdots,\sigma(\alpha_n))=\operatorname{rank}(A)$，故$\operatorname{dim}(I_m(\sigma))=\operatorname{rank}(A)$

(4)设$\operatorname{dim}(\operatorname{Ker}(\sigma))=r$，在$\operatorname{Ker}(\sigma)$中取一组基$\alpha_1,\alpha_2,\cdots,\alpha_r$并将其扩充到$V_1$的基$\alpha_1,\alpha_2,\cdots,\alpha_r,\alpha_{r+1},\cdots,\alpha_n$ 则有

由于$\sigma(\alpha_1),\sigma(\alpha_2),\cdots,\sigma(\alpha_r)=0$，则有

现在证明$\sigma(\alpha_{r+1}),\cdots,\sigma(\alpha_n)$是线性无关的

设$\sigma(\alpha_{r+1}),\cdots,\sigma(\alpha_n)$是线性相关的，则

则有

也就是说$\sum_{j=r+1}^{n}k_j\alpha_j\in \operatorname{Ker}(\sigma)$，而$\alpha_1,\alpha_2,\cdots,\alpha_r$是$\operatorname{Ker}(\sigma)$的一组基，故

而$\alpha_1,\alpha_2,\cdots,\alpha_r,\alpha_{r+1},\cdots,\alpha_n$线性无关，故只有$k_j=0$，与$\sigma(\alpha_{r+1}),\cdots,\sigma(\alpha_n)$是线性相关的矛盾

因此$\operatorname{dim}(I_m(\sigma))=n-r$，也就是有$\operatorname{dim}(I_m(\sigma))+\operatorname{dim}(\operatorname{Ker}(\sigma))=n$