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线性代数应该这样学（第三版）习题选做：3F

3F.6

设 \(V\) 是有限维的, \(v_1, ... ,v_m \in V\) . 定义线性映射 \(\Gamma : V' \to F^m\) 如下:

\[\Gamma(\varphi)=(\varphi(v_1), ..., \varphi(v_m))\]
  • a) 证明 \(v_1, ... ,v_m\) 张成 \(V\) 当且仅当 \(\Gamma\) 是单的.
  • b) 证明 \(v_1, ... ,v_m\) 是线性无关的当且仅当 \(\Gamma\) 是满的.

我们令 \(\dim V = n\) .

a)

  • \(v_1, ... ,v_m\) 张成 \(V\) \(\to\) \(\Gamma\) 是单的 :

不失一般性, 我们令 \(v_1, ... , v_n\) 是 \(V\) 的一组基. 令 \(\phi_i\) 是 \(v_i\) 的对偶基. 同时 \(v_i = \sum_{j=1}^n a_{i,j}v_j (i=n+1, ..., m)\)

假设存在 \(\Gamma(\varphi_1)=\Gamma(\varphi_2)\) . 令 \(\varphi_1 = \sum_{i=1}^n c_i \phi_i, \varphi_2 = \sum_{i=1}^n d_i \phi_i\) .

\[\begin{aligned} \Gamma(\varphi_1) &= (\sum_{i=1}^n c_i \phi_i(v_1), ..., \sum_{i=1}^n c_i \phi_i(v_n), \sum_{i=1}^n c_i \phi_i(\sum_{j=1}^n a_{n+1,j}v_j), ... , \sum_{i=1}^n c_i \phi_i(\sum_{j=1}^n a_{m,j}v_j)) \\ &= (c_1, ..., c_n, \sum_{i=1}^n a_{n+1,i}c_i, ... , \sum_{i=1}^n a_{m,i}c_i) \end{aligned}\]

同理

\[\Gamma(\varphi_2) = (d_1, ..., d_n, \sum_{i=1}^n a_{n+1,i}d_i, ... , \sum_{i=1}^n a_{m,i}d_i)\]

则有 \(c_i = d_i\) .

  • \(\Gamma\) 是单的 \(\to\) \(v_1, ... ,v_m\) 张成 \(V\) :

反证法, 假设 \(v_1, ... ,v_m\) 无法张成 \(V\) .

不失一般性, 我们令 \(v_1, ... , v_{n'}\) 是 \(v_i\) 的一组极大线性无关组, 同时它也可以扩展成 \(V\) 的一组基 \(w_i\). 令 \(\phi_i\) 是 \(w_i\) 的对偶基. 同时 \(v_i = \sum_{j=1}^{n'} a_{i,j}w_j (i=n'+1, ..., m)\)

则有 \(\forall \{c_i\}_{n'} \forall \varphi = \sum_{i=1}^{n'} c_i \phi_i + \sum_{i=n'+1}^{m}x_i \phi_i\)

\[\begin{aligned} \Gamma(\varphi) &= (\sum_{i=1}^n c_i \phi_i(v_1), ..., \sum_{i=1}^{n} c_i \phi_i(v_{n'}), \sum_{i=1}^n c_i \phi_i(\sum_{j=1}^n a_{n'+1,j}v_j), ... , \sum_{i=1}^n c_i \phi_i(\sum_{j=1}^n a_{m,j}v_j)) + (\sum_{i=n'+1}^{m} x_i \phi_i(v_1), ..., \sum_{i=n'+1}^{m} x_i \phi_i(v_{n'}), \sum_{i=n'+1}^{m} x_i \phi_i(\sum_{j=1}^{n'} a_{n'+1,j}v_j), ... , \sum_{i=n'+1}^{m} x_i \phi_i(\sum_{j=1}^{n'} a_{m,j}v_j)) \\ &= (c_1, ..., c_{n'}, \sum_{i=1}^{n'} a_{n'+1,i}c_i, ... , \sum_{i=1}^{n'} a_{m,i}c_i) + (0, ..., 0) \\ &= (c_1, ..., c_{n'}, \sum_{i=1}^{n'} a_{n'+1,i}c_i, ... , \sum_{i=1}^{n'} a_{m,i}c_i) \end{aligned}\]

因取值与 \(x_i\) 无关, 故其不为单射.

b) 的证明与 a) 类似.

3F.13

定义 \(T: \mathbb{R}^3 \to \mathbb{R}^2\) 为 \(T(x,y,z)=(4x+5y+6z,7x+8y+9z)\) . 设 \(\varphi_1 \varphi_2\) 是 \(\mathbb{R}^2\) 标准基的对偶基, \(\psi_1, \psi_2, \psi_3\) 是 \(\mathbb{R}^3\) 的标准基的对偶基.

  • a) 描述线性泛函 \(T'(\varphi_1)\) 和 \(T'(\varphi_2)\)
  • b) 将 \(T'(\varphi_1)\) 和 \(T'(\varphi_2)\) 写成 \(\psi_1, \psi_2, \psi_3\) 的线性组合

\(\varphi_1 = e_1' \to \varphi_1(x, y) = x\) .

\(\varphi_2 = e_2' \to \varphi_2(x, y) = y\) .

\(\psi_1 = e_1' \to \psi_1(x, y, z) = x\) .

\(\psi_2 = e_2' \to \psi_1(x, y, z) = y\) .

\(\psi_3 = e_3' \to \psi_1(x, y, z) = z\) .

a)

\[\begin{aligned} (T'(\varphi_1))(x,y,z) &= \varphi_1 \circ T (x,y,z) \\ &= \varphi_1(4x+5y+6z,7x+8y+9z) \\ &= 4x+5y+6z \end{aligned}\]
\[\begin{aligned} (T'(\varphi_2))(x,y,z) &= \varphi_2 \circ T (x,y,z) \\ &= \varphi_2(4x+5y+6z,7x+8y+9z) \\ &= 7x+8y+9z \end{aligned}\]

b)

\[\begin{aligned} (T'(\varphi_1))(x,y,z) &= 4x+5y+6z \\ &= 4\psi_1(x, y, z) + 5\psi_2(x, y, z) + 6\psi_3(x, y, z) \\ &= (4\psi_1 + 5\psi_2 + 6\psi_3) (x, y, z) \end{aligned}\]

即 \(T'(\varphi_1) = 4\psi_1 + 5\psi_2 + 6\psi_3\)

同理 \(T'(\varphi_2) = 7\psi_1 + 8\psi_2 + 9\psi_3\)

3F.27

设 \(T\in \mathcal{L}(\mathcal{P}_5(\mathbb{R}), \mathcal{P}_5(\mathbb{R}))\) 且 \(\mathrm{null} T' = \mathrm{span}(\varphi)\) , 这里 \(\varphi\) 是 \(\mathcal{P}_5(\mathbb{R})\) 上的由 \(\varphi(p)=p(8)\) 定义的线性泛函. 证明 \(\mathrm{range }T=\{ p \in \mathcal{P}_5(\mathbb{R}) : p(8)=0 \}\) .

由 3.107 , 有 \(\mathrm{null} T' = (\mathrm{range} T)^0\) .

则有 \((\mathrm{range} T)^0 = \mathrm{span}(\varphi)\) .

而根据零化子的定义

\[\begin{aligned} &(\mathrm{range} T)^0 = \{\varphi \in (\mathcal{P}_5(\mathbb{R}))' : \forall u \in \mathrm{range} T, \varphi(u) = 0\} \\ \Rightarrow &(\forall \phi \in \mathrm{span}(\varphi), \forall u \in \mathrm{range} T, \phi(u) = 0) \land (\forall \phi \notin \mathrm{span}(\varphi), \exists u \in \mathrm{range} T, \phi(u) \ne 0)\\ \Rightarrow &(\forall u \in \mathrm{range} T, u(8) = 0) \land (\forall u \notin \mathrm{range} T, u(8) \ne 0) \end{aligned}\]

即为题目所需证明.

Note

\(\forall \phi \in \mathrm{span}(\varphi), \forall p,\varphi(p)=0 \leftrightarrow \phi(p)=0\)

3F.37

设 \(U\) 是 \(V\) 的子空间, \(\pi : V \to V/U\) 是通常的商映射, 则 \(\pi' \in \mathcal{L}((V/U)', V')\) .

  • a) 证明 \(\pi'\) 是单的.
  • b) 证明 \(\mathrm{range }\pi'=U^0\) .
  • c) \(\pi'\) 是 \((V/U)'\) 到 \(U^0\) 的同构.

a)

由 3.107 , 有 \(\mathrm{null }\pi' = (\mathrm{range} \pi) ^ 0 = (V/U)^0\) , 而 \(\dim V/U + \dim (V/U)^0 =\dim V/U\) , 故 \(\dim (V/U)^0 = 0\) , 即 \(\mathrm{null }\pi' = \{0\}\) , 即为题目所需证明.

b)

由 3.109 , 有 \(\mathrm{range }\pi' = (\mathrm{null} \pi) ^ 0 = U^0\)

c)

由 a) 知 \(\pi'\) 是单的， 由 b) 知 \(\pi'\) 是满的 .