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张祖锦2023年数学专业真题分类70天之第60天

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发表于 2023-3-5 13:15:29 | 显示全部楼层 |阅读模式
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## 张祖锦2023年数学专业真题分类70天之第60天 --- 1358、 (4)、 在 $\displaystyle 4$ 维行向量空间 $\displaystyle \mathbb\{P\}^4$ 中, 记 $\displaystyle V\_1=L(\alpha\_1,\alpha\_2,\alpha\_3), V\_2=L(\beta\_1,\beta\_2)$, 其中 \begin\{aligned\} &\alpha\_1=(1,-1,1,1), \alpha\_2=(-2,1,2,-1), \alpha\_3=(-4,3,0,-3),\\\\ &\beta\_1=(1,2,1,0), \beta\_2=(4,2,-2,1). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 $\displaystyle V\_1+V\_2$ 的维数为 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$, $\displaystyle V\_1\cap V\_2$ 的维数为 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (安徽大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 由 \begin\{aligned\} &A=(\alpha\_1^\mathrm\{T\}, \alpha\_2^\mathrm\{T\}, \alpha\_3^\mathrm\{T\}, \beta\_1^\mathrm\{T\},\beta\_2^\mathrm\{T\})\\\\ =&\left(\begin\{array\}\{cccccccccccccccccccc\}1&-2&-4&1&4\\\\ -1&1&3&2&2\\\\ 1&2&0&1&-2\\\\ 1&-1&-3&0&1\end\{array\}\right) \to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&-2&0&-\frac\{1\}\{2\}\\\\ 0&1&1&0&-\frac\{3\}\{2\}\\\\ 0&0&0&1&\frac\{3\}\{2\}\\\\ 0&0&0&0&0\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle \alpha\_1,\alpha\_2,\beta\_1$ 线性无关, 且 [张祖锦注: 初等行变换不改变列向量组的秩及极大无关组所在的位置, 并且可以一下得到其余向量用极大无关组的表示法, 这是张祖锦独创的, 具体证明见张祖锦编著的《樊启斌参考书》中张祖锦常用的结论] \begin\{aligned\} \alpha\_3=-2\alpha\_1+\alpha\_2, \beta\_2=-\frac\{1\}\{2\}\alpha\_1-\frac\{3\}\{2\}\alpha\_2+\frac\{3\}\{2\}\beta\_1. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle \dim(V\_1+V\_2)=3$. 又由 $\displaystyle \dim V\_1=2, \dim V\_2=2$ 及维数公式知 $\displaystyle \dim(V\_1\cap V\_2)=1$.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1359、 (4)、 设 $\displaystyle V$ 为线性空间, $\displaystyle V\_1,V\_2,V\_3$ 为 $\displaystyle V$ 的子空间, 则 \begin\{aligned\} (V\_1+V\_2)\cap V\_3=(V\_1\cap V\_3)+(V\_2\cap V\_3) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 恒成立. (安徽大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / $\displaystyle \times$. 比如对 \begin\{aligned\} V\_1&=L\left((1,0,0), (0,1,0)\right),\\\\ V\_2&=L\left((1,0,0), (0,0,1)\right),\\\\ V\_3&=L\left((1,0,0), (0,1,1)\right), \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 有 \begin\{aligned\} (V\_1+V\_2)\cap V\_3&=V\_3,\\\\ (V\_1\cap V\_3)+(V\_2\cap V\_3)&=L\left((1,0,0)\right)+\left((1,0,0)\right) =L\left((1,0,0)\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1360、 4、 (20 分) 设 $\displaystyle V$ 是数域 $\displaystyle \mathbb\{F\}$ 上的 $\displaystyle n$ 维线性空间, $\displaystyle \alpha\_1,\cdots,\alpha\_n$ 为 $\displaystyle V$ 的一组基. 用 $\displaystyle V\_1$ 表示 $\displaystyle \alpha\_1+\cdots+\alpha\_n$ 生成的子空间. 令 \begin\{aligned\} V\_2=\left\\{\sum\_\{i=1\}^n k\_i\alpha\_i; \sum\_\{i=1\}^n k\_i=0, k\_i\in\mathbb\{F\}\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (1)、 证明: $\displaystyle V\_2$ 是 $\displaystyle V$ 的子空间, 且 $\displaystyle V=V\_1\oplus V\_2$; (2)、 设 $\displaystyle V$ 上的线性变换 $\displaystyle \varphi$ 在基 $\displaystyle \alpha\_1,\cdots,\alpha\_n$ 下的矩阵 $\displaystyle A$ 是置换矩阵 (即 $\displaystyle A$ 每行每列只有一个元素为 $\displaystyle 1$, 其余元素为 $\displaystyle 0$). 证明: $\displaystyle V\_1,V\_2$ 都是 $\displaystyle \varphi$ 的不变子空间. (北京工业大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 由 \begin\{aligned\} &\sum\_\{i=1\}^n k\_i\alpha\_i\in V\_2, \sum\_\{i=1\}^n l\_i\alpha\_i\in V\_2, \sum\_\{i=1\}^n k\_i=\sum\_\{i=1\}^n l\_i=0\\\\ \Rightarrow&\sum\_\{i=1\}^n (kk\_i+ll\_i) =k\sum\_\{i=1\}^n k\_i+l\sum\_\{i=1\}^n l\_i=k0+l0=0\\\\ \Rightarrow&k\left(\sum\_\{i=1\}^n k\_i\alpha\_i\right) +l\left(\sum\_\{i=1\}^n l\_i\alpha\_i\right) =\sum\_\{i=1\}^n (kk\_i+ll\_i)\alpha\_i \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 即知 $\displaystyle V\_2$ 是 $\displaystyle V$ 的子空间. (2)、 由 \begin\{aligned\} &V\ni \alpha=\sum\_\{i=1\}^n k\_i\alpha\_i =\frac\{\sum\_\{j=1\}^n k\_j\}\{n\}\sum\_\{i=1\}^n \alpha\_i+\sum\_\{i=1\}^n \left(k\_i-\frac\{\sum\_\{j=1\}^n k\_j\}\{n\}\right)\alpha\_i \in V\_1+V\_2,\\\\ &\alpha\in V\_1\cap V\_2 \Rightarrow \alpha=k\sum\_\{i=1\}^n \alpha\_i =\sum\_\{i=1\}^n k\_i\alpha\_i, \sum\_\{i=1\}^n k\_i=0\\\\ &\Rightarrow \sum\_\{i=1\}^n (k-k\_i)\alpha\_i=0 \Rightarrow k-k\_i=0\Rightarrow 0=\sum\_\{i=1\}^n k\_i=nk\\\\ &\Rightarrow k=0\Rightarrow \alpha=0 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 即知 $\displaystyle V=V\_1\oplus V\_2$. (3)、 由题设, 存在 $\displaystyle n$ 阶排列 $\displaystyle i\_1\cdots i\_n$ 使得 \begin\{aligned\} \varphi(\alpha\_j)=\alpha\_\{i\_j\}, 1\leq j\leq n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 从而 \begin\{aligned\} &\varphi(\alpha\_1+\cdots+\alpha\_n)=\varphi(\alpha\_1)+\cdots+\varphi(\alpha\_n)\\\\ =&\alpha\_\{i\_1\}+\cdots+\alpha\_\{i\_n\}=\alpha\_1+\cdots+\alpha\_n\in V\_1. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle V\_1$ 是 $\displaystyle \varphi$ 的不变子空间. 再者, 对 $\displaystyle \sum\_\{j=1\}^n k\_j=0$, \begin\{aligned\} \varphi\left(\sum\_\{j=1\}^n k\_j\alpha\_j\right) =\sum\_\{j=1\}^n k\_j\varphi(\alpha\_j) =\sum\_\{j=1\}^n k\_j\alpha\_\{i\_j\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 还是 $\displaystyle \alpha\_1,\cdots,\alpha\_n$ 的线性组合, 且组合系数的和还是 $\displaystyle 0$. 故 $\displaystyle \varphi\left(\sum\_\{j=1\}^n k\_j\alpha\_j\right)\in V\_2$. 这就证明了 $\displaystyle V\_2$ 也是 $\displaystyle \varphi$ 的不变子空间.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1361、 7、 (30 分) 设 $\displaystyle V$ 是数域 $\displaystyle \mathbb\{F\}$ 上的 $\displaystyle n$ 维线性空间, $\displaystyle \sigma\_1,\cdots,\sigma\_s$ 都是 $\displaystyle V$ 上的非零线性变换, $\displaystyle \mathrm\{rank\} \sigma$ 表示线性变换 $\displaystyle \sigma$ 的秩. (1)、 证明: 存在 $\displaystyle \alpha\in V$, 使得 $\displaystyle \sigma\_i(\alpha)\neq 0, i=1,\cdots,s$; (2)、 令 $\displaystyle \sigma=\sigma\_1+\cdots+\sigma\_s$. 证明: $\displaystyle \sigma$ 是幂等变换且 $\displaystyle \mathrm\{rank\} \sigma=\sum\_\{i=1\}^s \mathrm\{rank\} \sigma\_i$ 的充要条件为 $\displaystyle \sigma\_1,\cdots,\sigma\_s$ 都是幂等变换, 且 $\displaystyle \sigma\_i\sigma\_j=0\ (i\neq j)$ (北京工业大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 先证明一个结论: 非完全覆盖原理 (线性空间不能被其任意有限个真子空间覆盖). 设 $\displaystyle V$ 是 $\displaystyle n$ 维线性空间, $\displaystyle V\_1,\cdots,V\_m$ 是 $\displaystyle V$ 的 $\displaystyle m$ 个真子空间, 则 \begin\{aligned\} \exists\ \mbox\{无限多个\}\alpha\in\left\\{\beta\_j\right\\}\_\{j=1\}^\infty\subset V,\mathrm\{ s.t.\} \alpha\not\in V\_1\cup \cdots \cup V\_m. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 事实上, 取定 $\displaystyle V$ 的一组基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$, 令 \begin\{aligned\} \alpha\_i=\varepsilon\_1+i \varepsilon\_2+\cdots+i^\{n-1\}\varepsilon\_n, i=1,2,\cdots, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 $\displaystyle \alpha\_1,\alpha\_2,\cdots$ 中的任意 $\displaystyle n$ 个 \begin\{aligned\} (\alpha\_\{i\_1\},\cdots,\alpha\_\{i\_n\})=(\varepsilon\_1,\cdots,\varepsilon\_n)\left(\begin\{array\}\{cccccccccccccccccccc\} 1&1&\cdots&1\\\\ i\_1&i\_2&\cdots&i\_n\\\\ \vdots&\vdots&&\vdots\\\\ i\_1^\{n-1\}&i\_2^\{n-1\}&\cdots&i\_n^\{n-1\}\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 由范德蒙行列式知 $\displaystyle \alpha\_\{i\_1\},\cdots,\alpha\_\{i\_n\}$ 是 $\displaystyle V$ 的一组基. 由于 $\displaystyle V\_i$ 是 $\displaystyle V$ 的真子空间, 而每个 $\displaystyle V\_i$ 至多包含 $\displaystyle \alpha\_1,\alpha\_2,\cdots$ 中 的 $\displaystyle n-1$ 个, $\displaystyle \bigcup\_\{i=1\}^m V\_i$ 至多包含 $\displaystyle \alpha\_1,\alpha\_2,\cdots$ 中的 $\displaystyle m(n-1)$ 个. 于是 \begin\{aligned\} \exists\ \mbox\{无限多个\} k,\mathrm\{ s.t.\} \alpha\_k\not\in \bigcup\_\{i=1\}^mV\_i. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (2)、 回到题目. 由 $\displaystyle \sigma\_i$ 非零知 $\displaystyle \ker\sigma\_i$ 是 $\displaystyle V$ 的真子空间, 而由第 1 步知 \begin\{aligned\} \exists\ \alpha\in V,\mathrm\{ s.t.\} \alpha\not\in \bigcup\_\{i=1\}^s \ker\sigma\_i \Rightarrow\forall\ 1\leq i\leq s, \sigma\_i(\alpha)\neq 0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (3)、 为证第 2 问, 我们给出一个结论. 设 $\displaystyle A$ 为 $\displaystyle n$ 阶幂等矩阵 (即 $\displaystyle A^2=A$). (3-1)、 证明 $\displaystyle A$ 相似于对角阵, 且 $\displaystyle \mathrm\{rank\} A=\mathrm\{tr\} A$; (3-2)、 $\displaystyle \mathrm\{rank\} A+\mathrm\{rank\}(I-A)=n$, 其中 $\displaystyle I$ 为 $\displaystyle n$ 阶单位阵. 事实上, (3-1)、 设 \begin\{aligned\} V\_1=\left\\{x; Ax=x\right\\}, V\_2=\left\\{x; Ax=0\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 \begin\{aligned\} &\mathbb\{P\}^n\ni \alpha=A\alpha+(\alpha-A\alpha)\ni V\_1+V\_2,\\\\ &\alpha\in V\_1\cap V\_2\Rightarrow A\alpha=\alpha, A\alpha=0\Rightarrow \alpha=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle \mathbb\{P\}^n=V\_1\oplus V\_2$. 取定 $\displaystyle V\_1$ 的一组基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_r$, $\displaystyle V\_2$ 的一组基 $\displaystyle \varepsilon\_\{r+1\},\cdots,\varepsilon\_n$, 则 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$ 是 $\displaystyle V$ 的一组基. 令 $\displaystyle P=(\alpha\_1,\cdots,\alpha\_n)$, 则 \begin\{aligned\} AP=P\left(\begin\{array\}\{cccccccccccccccccccc\}E\_r&\\\\ &0\end\{array\}\right)\Rightarrow P^\{-1\}AP=\left(\begin\{array\}\{cccccccccccccccccccc\}E\_r&\\\\ &0\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (3-2)、 由 \begin\{aligned\} \left(\begin\{array\}\{cccccccccccccccccccc\}A&\\\\ &A-E\end\{array\}\right)\to&\left(\begin\{array\}\{cccccccccccccccccccc\}A&A\\\\ &A-E\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}A&A\\\\ -A&-E\end\{array\}\right)\\\\ \to&\left(\begin\{array\}\{cccccccccccccccccccc\}A-A^2&\\\\ -A&-E\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}A-A^2&\\\\ &E\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 \begin\{aligned\} \mathrm\{rank\}(A)+\mathrm\{rank\}(A-E)=\mathrm\{rank\}(A-A^2)+n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 \begin\{aligned\} \mathrm\{rank\}(A)+\mathrm\{rank\}(A-E)=n&\Leftrightarrow \mathrm\{rank\}(A-A^2)=0\\\\ &\Leftrightarrow A-A^2=0\Leftrightarrow A=A^2. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (4)、 取定 $\displaystyle V$ 的一组基, 设 $\displaystyle \sigma\_i,\sigma$ 在该基下的矩阵分别为 $\displaystyle A\_i,A$, 则 $\displaystyle A=\sum\_\{i=1\}^s A\_i$, 且只要证明 \begin\{aligned\} A^2=A, \mathrm\{rank\} A=\sum\_\{i=1\}^s \mathrm\{rank\} A\_i \Leftrightarrow A\_i^2=A\_i, A\_iA\_j=0, \forall\ 1\leq i\neq j\leq n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (4-1)、 $\displaystyle \Leftarrow$: \begin\{aligned\} A^2=\left(\sum\_\{i=1\}^s A\_i\right)^2=\sum\_\{i=1\}^s A\_i \sum\_\{j=1\}^s A\_j =\sum\_\{i=1\}^s A\_i^2=\sum\_\{i=1\}^s A\_i=A, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 且 \begin\{aligned\} \mathrm\{rank\} A=\mathrm\{tr\} A=\mathrm\{tr\}\left(\sum\_\{i=1\}^s A\_i\right)=\sum\_\{i=1\}^s \mathrm\{tr\} A\_i=\sum\_\{i=1\}^s \mathrm\{rank\} A\_i. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (4-2)、 $\displaystyle \Rightarrow$: 由 \begin\{aligned\} &\left(\begin\{array\}\{cccccccccccccccccccc\}A\_1&&&\\\\ &\ddots&&\\\\ &&A\_s&\\\\ &&&I-A\end\{array\}\right) \to \left(\begin\{array\}\{cccccccccccccccccccc\}A\_1&&&\\\\ &\ddots&&\\\\ &&A\_s&\\\\ A\_1&\cdots&A\_s&I-A\end\{array\}\right)\\\\ \to& \left(\begin\{array\}\{cccccccccccccccccccc\}A\_1&&&A\_1\\\\ &\ddots&&A\_s\\\\ &&A\_s&\vdots\\\\ A\_1&\cdots&A\_s&I\end\{array\}\right) \to \left(\begin\{array\}\{cccccccccccccccccccc\}A\_1-A\_1^2&-A\_1A\_2&\cdots&-A\_1A\_s&0\\\\ -A\_2A\_1&A\_2-A\_2^2&\cdots&-A\_2A\_s&0\\\\ \vdots&\vdots&&\vdots&\vdots\\\\ -A\_sA\_1&-A\_sA\_2&\cdots&A\_s-A\_s^2&0\\\\ A\_1&A\_2&\cdots&A\_s&I\end\{array\}\right)\\\\ \to&\left(\begin\{array\}\{cccccccccccccccccccc\}A\_1-A\_1^2&-A\_1A\_2&\cdots&-A\_1A\_s&0\\\\ -A\_2A\_1&A\_2-A\_2^2&\cdots&-A\_2A\_s&0\\\\ \vdots&\vdots&&\vdots&\vdots\\\\ -A\_sA\_1&-A\_sA\_2&\cdots&A\_s-A\_s^2&0\\\\ 0&0&\cdots&0&I\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 \begin\{aligned\} &\sum\_\{i=1\}^s \mathrm\{rank\} A\_i+\mathrm\{rank\}(I-A)\\\\ =&n+\mathrm\{rank\}\left(\begin\{array\}\{cccccccccccccccccccc\}A\_1-A\_1^2&-A\_1A\_2&\cdots&-A\_1A\_s\\\\ -A\_2A\_1&A\_2-A\_2^2&\cdots&-A\_2A\_s\\\\ \vdots&\vdots&&\vdots&\\\\ -A\_sA\_1&-A\_sA\_2&\cdots&A\_s-A\_s^2\end\{array\}\right).\qquad(I) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 由 $\displaystyle A^2=A$ 知 $\displaystyle \mathrm\{rank\}(I-A)+\mathrm\{rank\} A=n$. 联合 $\displaystyle (I)$ 及 $\displaystyle \mathrm\{rank\} A=\sum\_\{i=1\}^s \mathrm\{rank\} A\_i$ 知 \begin\{aligned\} \mathrm\{rank\}\left(\begin\{array\}\{cccccccccccccccccccc\}A\_1-A\_1^2&-A\_1A\_2&\cdots&-A\_1A\_s\\\\ -A\_2A\_1&A\_2-A\_2^2&\cdots&-A\_2A\_s\\\\ \vdots&\vdots&&\vdots&\\\\ -A\_sA\_1&-A\_sA\_2&\cdots&A\_s-A\_s^2\end\{array\}\right)=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 从而 $\displaystyle A\_i=A\_i^2, A\_iA\_j=0\left(i\neq j\right)$. 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1362、 3、 (20 分) 已知矩阵 \begin\{aligned\} A=\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&0\\\\ -1&2&1\\\\ 0&1&1\\\\ -2&6&4\end\{array\}\right), \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} $\displaystyle \mathbb\{R\}^\{n\times 5\}$ 为实数域 $\displaystyle \mathbb\{R\}$ 上的所有 $\displaystyle n\times 5$ 矩阵构成的线性空间, \begin\{aligned\} W=\left\\{B\in\mathbb\{R\}^\{n\times 5\}; BA=0\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 其中 $\displaystyle 0$ 为零矩阵. (1)、 证明: $\displaystyle W$ 是 $\displaystyle \mathbb\{R\}^\{n\times 5\}$ 的子空间; (2)、 求 $\displaystyle W$ 的一组基和维数. [题目有问题, 跟锦数学微信公众号没法做哦.] (北京科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / [题目有问题, 跟锦数学微信公众号没法做哦.]跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1363、 5、 (20 分) $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 为数域 $\displaystyle \mathbb\{P\}$ 上的 $\displaystyle 2\times 2$ 方阵构成的线性空间. 令 $\displaystyle \sigma: \mathbb\{P\}^\{2\times 2\}\to \mathbb\{P\}^\{2\times 2\}$, \begin\{aligned\} \sigma(X)=AXB, \forall\ X\in\mathbb\{P\}^\{2\times 2\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 其中 $\displaystyle A=\left(\begin\{array\}\{cccccccccccccccccccc\}-1&1\\\\ 1&-1\end\{array\}\right), B=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ 1&0\end\{array\}\right)$. (1)、 证明: $\displaystyle \sigma$ 是 $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 上的线性变换; (2)、 求 $\displaystyle \sigma$ 在 $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 的基 \begin\{aligned\} E\_\{11\}=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0\\\\ 0&0\end\{array\}\right), E\_\{12\}=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ 0&0\end\{array\}\right), E\_\{21\}=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0\\\\ 1&0\end\{array\}\right), E\_\{22\}=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0\\\\ 0&1\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 下的表示矩阵; (3)、 是否存在 $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 的某组基, 使得 $\displaystyle \sigma$ 在此基下的矩阵为对角阵? 存在的话, 求出基和对应的对角阵. (北京科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 对 $\displaystyle \forall\ k,l\in\mathbb\{P\}, X,Y\in \mathbb\{P\}^\{2\times 2\}$, 由 \begin\{aligned\} \sigma(kX+lY)&=A(kX+lY)B=kAXB+lAYB=k\sigma(X)+l\sigma(Y) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle \sigma$ 是 $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 上的线性变换. (2)、 由 \begin\{aligned\} &\sigma(E\_\{11\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&-1\\\\0&1\end\{array\}\right), \sigma(E\_\{12\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\0&0\end\{array\}\right),\\\\ &\sigma(E\_\{21\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0\\\\1&0\end\{array\}\right), \sigma(E\_\{22\})=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0\\\\-1&0\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 \begin\{aligned\} \sigma(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\})=(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\})C, C=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0&0&1\\\\ -1&1&0&0\\\\ 0&0&1&-1\\\\ 1&0&0&0\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (3)、 易知 $\displaystyle C$ 的特征值为 $\displaystyle 1,1,1,-1$. 由 \begin\{aligned\} C-E\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&0\\\\ 0&0&0&1\\\\ 0&0&0&0\\\\ 0&0&0&0 \end\{array\}\right)\Rightarrow \mathrm\{rank\}(C-E)=2. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle C$ 的 Jordan 标准形 $\displaystyle J$ 满足 $\displaystyle \mathrm\{rank\}(J-E)=2$, 而 $\displaystyle J$ 的严格上三角部分有一个 $\displaystyle 1$, \begin\{aligned\} J=\left(\begin\{array\}\{cccccccccccccccccccc\}1&1&&\\\\ &1&&\\\\ &&-1&\\\\ &&&-1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 这表明 $\displaystyle \sigma$ 不可对角化, 即不存在 $\displaystyle \mathbb\{P\}^\{2\times 2\}$ 的某组基, 使得 $\displaystyle \sigma$ 在此基下的矩阵为对角阵跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1364、 (2)、 向量组 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3$ 的秩为 $\displaystyle 3$, 则 \begin\{aligned\} \alpha\_1+\alpha\_2, \alpha\_2+\alpha\_3, \alpha\_3+\alpha\_1 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 的秩为 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (北京理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (2-1)、 我们给出一般的结果. 已知向量组 $\displaystyle \alpha\_1,\cdots,\alpha\_s\ (s > 1)$ 线性无关, 设 \begin\{aligned\} \beta\_1=\alpha\_1+\alpha\_2, \beta\_2=\alpha\_2+\alpha\_3, \cdots, \beta\_s=\alpha\_s+\alpha\_1. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 讨论向量 $\displaystyle \beta\_1,\cdots, \beta\_s$ 的线性相关性. 写出 \begin\{aligned\} (\beta\_1,\cdots,\beta\_s)=(\alpha\_1,\cdots,\alpha\_s)A, A=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&\cdots&1\\\\ 1&1&\cdots&0\\\\ 0&1&\cdots&0\\\\ \vdots&\vdots&&\vdots\\\\ 0&0&\cdots&1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 按第一行展开, \begin\{aligned\} |A|=1+(-1)^\{1+s\}=\left\\{\begin\{array\}\{llllllllllll\}2\neq 0,&s\mbox\{为奇数\},\\\\ 0,&s\mbox\{为偶数\}.\end\{array\}\right. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故当 $\displaystyle s$ 为奇数时, $\displaystyle \beta\_1,\cdots,\beta\_s$ 线性无关; 当 $\displaystyle s$ 为偶数时, $\displaystyle \beta\_1,\cdots,\beta\_s$ 线性相关. (2-2)、 回到题目. 由第 i 步知 \begin\{aligned\} \alpha\_1+\alpha\_2, \alpha\_2+\alpha\_3, \alpha\_3+\alpha\_1 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 线性无关.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1365、 (3)、 已知 \begin\{aligned\} \alpha\_1=(1,-1,0)^\mathrm\{T\}, \alpha\_2=(2,1,3)^\mathrm\{T\}, \alpha\_3=(3,1,2)^\mathrm\{T\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 是 $\displaystyle \mathbb\{R\}^3$ 的一组基, 则 $\displaystyle \beta=(1,1,1)^\mathrm\{T\}$ 在这组基下的坐标为 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (北京理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设 \begin\{aligned\} &\beta=x\_1\alpha\_1+x\_2\alpha\_2+x\_3\alpha\_3\Leftrightarrow \beta=(\alpha\_1,\alpha\_2,\alpha\_3)X\\\\ \Leftrightarrow&X=(\alpha\_1,\alpha\_2,\alpha\_3)^\{-1\}\beta=\left(\begin\{array\}\{cccccccccccccccccccc\}-\frac\{1\}\{2\}\\\\0\\\\\frac\{1\}\{2\}\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1366、 4、 写出数域 $\displaystyle \mathbb\{P\}$ 上线性空间 $\displaystyle V$ 上的线性变换 $\displaystyle \sigma$ 可对角化的充要条件, 至少四个. (北京理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 $\displaystyle \sigma$ 可对角化当且仅当存在 $\displaystyle \sigma$ 有 $\displaystyle n$ 个线性无关的特征向量; (2)、 $\displaystyle \sigma$ 可对角化当且仅当存在 $\displaystyle \sigma$ 的特征子空间的维数之和为 $\displaystyle \dim V$; (3)、 $\displaystyle \sigma$ 可对角化当且仅当 $\displaystyle \sigma$ 的每个特征值都在 $\displaystyle \mathbb\{P\}$ 中, 且它的代数重数等于几何重数; (4)、 $\displaystyle \sigma$ 可对角化当且仅当存在 $\displaystyle \sigma$ 的最小多项式在 $\displaystyle \mathbb\{P\}$ 中的因式分解中因式全都是一次的.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1367、 4、 设 $\displaystyle \alpha\_1,\cdots,\alpha\_n$ 是秩为 $\displaystyle r$ 的向量组, \begin\{aligned\} J=\left\\{(k\_1,\cdots,k\_n)\in\mathbb\{R\}^n; \sum\_\{i=1\}^n k\_i\alpha\_i=0\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 证明: $\displaystyle \dim J=n-r$. (北京师范大学2023年高等代数与解析几何考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设 $\displaystyle A=(\alpha\_1,\cdots,\alpha\_n)$, 则 $\displaystyle \mathrm\{rank\} A=r$. 由 \begin\{aligned\} x=(k\_1,\cdots,k\_n)\in J\Leftrightarrow&\sum\_\{i=1\}^n k\_i\alpha\_i=0\Leftrightarrow Ax^\mathrm\{T\}=0 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle J\ni x\mapsto x^\mathrm\{T\}\in \left\\{y; Ay=0\right\\}$ 是同构映射, 从而 \begin\{aligned\} \dim J=\dim\left\\{y; Ay=0\right\\}=n-\mathrm\{rank\} A=n-r. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1368、 4、 设向量组 \begin\{aligned\} &\alpha\_1=(1,1,1,3)^\mathrm\{T\}, \alpha\_2=(-1,-3,5,1)^\mathrm\{T\},\\\\ &\alpha\_3=(3,2,-1,p+2)^\mathrm\{T\}, \alpha\_4=(-2,-6,10,p)^\mathrm\{T\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (1)、 $\displaystyle p$ 为何值时, 该向量组线性无关? 并将向量 $\displaystyle \alpha=(4,1,6,10)^\mathrm\{T\}$ 用 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4$ 线性表出; (2)、 $\displaystyle p$ 为何值时, 该向量组线性相关? 并求出它的秩和一个极大线性无关组. (北京邮电大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / $\displaystyle (\alpha\_1,\cdots,\alpha\_4,\alpha)$ \begin\{aligned\} =&\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&3&-2&4\\\\ 1&-3&2&-6&1\\\\ 1&5&-1&10&6\\\\ 3&1&p=2&p&10\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&3&-2&4\\\\ 0&-2&-1&-4&-3\\\\ 0&6&-4&12&2\\\\ 0&4&p-7&p+6&-2\end\{array\}\right)\\\\ \to&\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&3&-2&4\\\\ 0&-2&-1&-4&-3\\\\ 0&0&-7&0&-7\\\\ 0&0&p-9&p-2&-8\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&3&-2&4\\\\ 0&-2&0&-4&-2\\\\ 0&0&1&0&1\\\\ 0&0&0&p-2&1-p\end\{array\}\right)\\\\ \to&\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&0&2\\\\ 0&1&0&2&1\\\\ 0&0&1&0&1\\\\ 0&0&0&p-2&1-p\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 (1)、 当$p\neq 2$ 时, $\displaystyle \alpha\_1,\cdots,\alpha\_4$ 线性无关, 且上式可简单化简后知 \begin\{aligned\} \alpha=2\alpha\_1+\frac\{3p-4\}\{p-2\}\alpha\_2+\alpha\_3+\frac\{1-p\}\{p-2\}\alpha\_4. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (2)、 当 $\displaystyle p=2$ 时, $\displaystyle \alpha\_1,\cdots,\alpha\_4$ 线性相关, 且它的秩为 $\displaystyle 3$, $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3$ 是它的一个极大线性无关组.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1369、 8、 设 $\displaystyle \sigma$ 是线性空间 $\displaystyle V$ 上的线性变换, $\displaystyle \sigma$ 在基 $\displaystyle \alpha\_1,\cdots,\alpha\_4$ 下的矩阵是 \begin\{aligned\} \left(\begin\{array\}\{cccccccccccccccccccc\}1&-1&-1&2\\\\ 0&1&0&0\\\\ 2&3&1&-1\\\\ 1&-2&-2&-1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 求包含 $\displaystyle \alpha\_1$ 的最小的 $\displaystyle \sigma$ 的不变子空间. (北京邮电大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设题中矩阵为 $\displaystyle A$, 则 \begin\{aligned\} &(\alpha\_1,\sigma\alpha\_1,\sigma^2\alpha\_1,\sigma^3\alpha\_1)=(\alpha\_1,\cdots,\alpha\_4)B,\\\\ &B=\left(\begin\{array\}\{cccccccccccccccccccc\}e\_1,Ae\_1,A^2e\_1,A^3e\_1\end\{array\}\right)=\left(\begin\{array\}\{cccccccccccccccccccc\}1&1&1&-10\\\\ 0&0&0&0\\\\ 0&2&3&4\\\\ 0&1&-4&-1\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&-14\\\\ 0&1&0&3\\\\ 0&0&1&1\\\\ 0&0&0&0\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故包含 $\displaystyle \alpha\_1$ 的最小的 $\displaystyle \sigma$ 的不变子空间为 $\displaystyle L(\alpha\_1,\sigma\alpha\_1,\sigma^2\alpha\_1)$, 且 \begin\{aligned\} \sigma^3\alpha\_1=-14\alpha\_1+3\sigma\alpha\_1+\sigma^2\alpha\_1. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1370、 10、 证明: 数域 $\displaystyle \mathbb\{P\}$ 上的一元多项式组成的线性空间 $\displaystyle \mathbb\{P\}[x]$ 可以与它的一个真子空间同构. (北京邮电大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设 $\displaystyle \sigma: \mathbb\{P\}[x]\to\mathbb\{P\}[x]$ 为 $\displaystyle \sigma\left\[f(x)\right\]=f(x^2)$, 则由 \begin\{aligned\} \sigma\left\[kf(x)+lg(x)\right\]=kf(x^2)+lg(x^2)=k\sigma\left\[f(x)\right\]+l\sigma\left\[g(x)\right\] \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle \sigma$ 是线性变换. 又由 $\displaystyle \sigma\left\[f(x)\right\]=f(x^2)=0\Rightarrow f(x)=0$, $\displaystyle x\in\mathbb\{P\}[x]\backslash \mathrm\{im\} \sigma$ 知 $\displaystyle \sigma$ 是 $\displaystyle \mathbb\{P\}[x]$ 到它的一个真子空间 $\displaystyle \mathrm\{im\} \sigma$ 的一个同构.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1371、 (4)、 设 $\displaystyle \mathscr\{A\}$ 是 $\displaystyle n$ 维线性空间 $\displaystyle V$ 上的线性变换, 用 $\displaystyle \mathrm\{im\} \mathscr\{A\}$ 和 $\displaystyle \ker \mathscr\{A\}$ 分别表示 $\displaystyle \mathscr\{A\}$ 的值域和核. 证明: $\displaystyle \mathrm\{im\}\mathscr\{A\}\subset \ker \mathscr\{A\}$ 的充要条件是 $\displaystyle \mathscr\{A\}^2$ 等于零变换. (大连理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (4-1)、 $\displaystyle \Rightarrow$: 对 $\displaystyle \forall\ \alpha\in V$, \begin\{aligned\} \mathscr\{A\}\alpha\in \mathrm\{im\} \mathscr\{A\}\subset \ker \mathscr\{A\}\Rightarrow \mathscr\{A\}^2\alpha=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle \mathscr\{A\}^2=\mathscr\{O\}$. (4-2)、 $\displaystyle \Leftarrow$: 设 $\displaystyle \alpha\in\mathrm\{im\}\mathscr\{A\}$, 则 \begin\{aligned\} \exists\ \beta\in V,\mathrm\{ s.t.\} \alpha=\mathscr\{A\}\beta \Rightarrow \mathscr\{A\}\alpha=\mathscr\{A\}^2\beta=\mathscr\{O\}\beta=0\Rightarrow \alpha\in \ker \mathscr\{A\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle \mathrm\{im\}\mathscr\{A\}\subset \ker \mathscr\{A\}$.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1372、 (5)、 设 $\displaystyle A$ 是数域 $\displaystyle \mathbb\{P\}$ 上的 $\displaystyle m\times n$ 矩阵, $\displaystyle B$ 是数域 $\displaystyle \mathbb\{P\}$ 上的 $\displaystyle (n-m)\times n$ 矩阵. 令 \begin\{aligned\} V\_1=&\left\\{X\in\mathbb\{P\}^n; AX=0\right\\},\\\\ V\_2=&\left\\{X\in\mathbb\{P\}^n; BX=0\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 已知矩阵 $\displaystyle C=\left(\begin\{array\}\{cccccccccccccccccccc\}A\\\\B\end\{array\}\right)$ 可逆. 证明: $\displaystyle \mathbb\{P\}^n=V\_1\oplus V\_2$. (大连理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 由 $\displaystyle \left(\begin\{array\}\{cccccccccccccccccccc\}A\\\\B\end\{array\}\right)$ 非奇异知 \begin\{aligned\} X\in V\_1\cap V\_2&\Rightarrow \left(\begin\{array\}\{cccccccccccccccccccc\}A\\\\B\end\{array\}\right)X=0\Rightarrow X=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle V\_1\cap V\_2=\left\\{0\right\\}$. 又由 $\displaystyle \left(\begin\{array\}\{cccccccccccccccccccc\}A\\\\B\end\{array\}\right)$ 非奇异知 \begin\{aligned\} &\mathrm\{rank\}(A)=r, \mathrm\{rank\}(B)=n-r\Rightarrow \dim V\_1=n-r, \dim V\_2=r\\\\ \Rightarrow&\dim (V\_1+V\_2)=\dim V\_1+\dim V\_2-\dim(V\_1\cap V\_2)\\\\ &=(n-r)+r-0=n\\\\ \Rightarrow&V\_1+V\_2=\mathbb\{P\}^n\Rightarrow V\_1\oplus V\_2=\mathbb\{P\}^n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1373、 (6)、 设 $\displaystyle \mathscr\{A\}$ 是 $\displaystyle n$ 维线性空间 $\displaystyle V$ 上的线性变换, $\displaystyle W$ 是 $\displaystyle V$ 的 $\displaystyle \mathscr\{A\}$-子空间. 已知 $\displaystyle \mathscr\{A\}$ 有 $\displaystyle k$ 个互异特征值 $\displaystyle \lambda\_1,\cdots,\lambda\_k$, 相应的特征向量分别为 $\displaystyle \xi\_1,\cdots,\xi\_k$. 证明: 若 $\displaystyle \xi\_1+\cdots+\xi\_k\in W$, 则 $\displaystyle \dim W\geq k$. (大连理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设 $\displaystyle \xi\_1+\cdots+\xi\_k=\alpha\in W$, 则 \begin\{aligned\} \lambda\_1\xi\_1+\cdots+\lambda\_k\xi\_k=\mathscr\{A\}\alpha\in W, \cdots, \lambda\_1^\{k-1\}\xi\_1+\cdots+\lambda\_k^\{k-1\}\xi\_k=\mathscr\{A\}^\{k-1\}\alpha\in W. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 于是 \begin\{aligned\} &(\xi\_1,\cdots,\xi\_k)\left(\begin\{array\}\{cccccccccccccccccccc\}1&\lambda\_1&\cdots&\lambda\_1^\{k-1\}\\\\ 1&\lambda\_2&\cdots&\lambda\_2^\{k-1\}\\\\ \vdots&\vdots&&\vdots\\\\ 1&\lambda\_k&\cdots&\lambda\_k^\{k-1\}\end\{array\}\right)=(\alpha,\mathscr\{A\}\alpha,\cdots,\mathscr\{A\}^\{k-1\}\alpha)\\\\ \Rightarrow&(\xi\_1,\cdots,\xi\_k)=(\alpha,\mathscr\{A\}\alpha,\cdots,\mathscr\{A\}^\{k-1\}\alpha) \left(\begin\{array\}\{cccccccccccccccccccc\}1&\lambda\_1&\cdots&\lambda\_1^\{k-1\}\\\\ 1&\lambda\_2&\cdots&\lambda\_2^\{k-1\}\\\\ \vdots&\vdots&&\vdots\\\\ 1&\lambda\_k&\cdots&\lambda\_k^\{k-1\}\end\{array\}\right)^\{-1\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 这表明 $\displaystyle \xi\_1,\cdots,\xi\_k$ 是 $\displaystyle \alpha,\mathscr\{A\}\alpha,\cdots,\mathscr\{A\}^\{k-1\}\alpha\in W$ 的线性组合, 而 $\displaystyle \xi\_1,\cdots,\xi\_k\in W$. 又由 $\displaystyle \xi\_1,\cdots,\xi\_k$ 是 $\displaystyle \mathscr\{A\}$ 属于不同特征值的特征向量知它们线性无关. 于是 \begin\{aligned\} \dim W\geq \dim L\left(\xi\_1,\cdots,\xi\_k\right)=k. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1374、 (2)、 已知矩阵 \begin\{aligned\} A=\left(\begin\{array\}\{cccccccccccccccccccc\}a\_\{11\}&a\_\{12\}&\cdots&a\_\{1n\}\\\\ a\_\{21\}&a\_\{22\}&\cdots&a\_\{2n\}\\\\ \vdots&\vdots&&\vdots\\\\ a\_\{n1\}&a\_\{n2\}&\cdots&a\_\{nn\}\end\{array\}\right), F=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0&\cdots&0&-a\_n\\\\ 1&0&\cdots&0&-a\_\{n-1\}\\\\ 0&1&\cdots&0&-a\_\{n-2\}\\\\ \vdots&\vdots&&\vdots&\vdots\\\\ 0&0&\cdots&1&-a\_1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (2-1)、 若 $\displaystyle AF=FA$, 证明: \begin\{aligned\} A=a\_\{11\}E+a\_\{21\}F+a\_\{31\}F^2+\cdots+a\_\{n1\}F^\{n-1\}; \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (2-2)、 求子空间 $\displaystyle C(F)=\left\\{B\in\mathbb\{C\}^\{n\times n\};BF=BF\right\\}$ 的维数. (大连理工大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (2-1)、 设 \begin\{aligned\} \beta&=(-a\_n,-a\_\{n-1\},\cdots,-a\_1)^\mathrm\{T\},\\\\ e\_i&=(\underbrace\{0,\cdots,0,1\}\_i,0,\cdots,0)^\mathrm\{T\},\\\\ M&=a\_\{n1\}F^\{n-1\}+a\_\{n-1,1\}F^\{n-2\}+\cdots+a\_\{21\}F+a\_\{11\}E, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 \begin\{aligned\} F=(e\_2,\cdots,e\_n,\beta), MF^i=F^iM\left(\forall\ i\in\mathbb\{N\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 为证 \begin\{aligned\} A=M\Leftrightarrow Ae\_i=Me\_i\left(1\leq i\leq n\right), \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 我们一个一个去验算 (从第一个就可看出 $\displaystyle M$ 的形状): \begin\{aligned\} Ae\_1&=\left(\begin\{array\}\{cccccccccccccccccccc\}a\_\{11\}\\\\a\_\{21\}\\\\\vdots\\\\a\_\{n1\}\end\{array\}\right)=a\_\{11\}e\_1+a\_\{21\}e\_2+\cdots+a\_\{n1\}e\_n\\\\ &=a\_\{11\}Ee\_1+a\_\{21\}Fe\_1+\cdots+a\_\{n1\}F^\{n-1\}e\_1\\\\ &=Me\_1,\\\\ Ae\_2&=AFe\_1=FAe\_1=FMe\_1=MFe\_1=Me\_2,\\\\ \cdot&=\cdots,\\\\ A\_n&=AF^\{n-1\}e\_1=F^\{n-1\}Ae\_1=F^\{n-1\}Me\_1=MF^\{n-1\}e\_1=Me\_n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (2-2)、 由第 1 步即知 \begin\{aligned\} C(F)=L(E,F,F^2,\cdots,F^\{n-1\}). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 往证 $\displaystyle E,F,F^2,\cdots,F^\{n-1\}$ 线性无关, 而 $\displaystyle \dim C(F)=n$. 事实上, \begin\{aligned\} &k\_0E+k\_1F+k\_2F^2+\cdots+k\_\{n-1\}F^\{n-1\}=0\\\\ \Rightarrow&k\_0Ee\_1+k\_1Fe\_1+k\_2F^2e\_1+\cdots+k\_\{n-1\}F^\{n-1\}e\_1=0\\\\ \Rightarrow&k\_0e\_1+k\_1e\_2+k\_2e\_3+\cdots+k\_\{n-1\}e\_n=0\\\\ \Rightarrow&k\_0=k\_1=\cdots=k\_\{n-1\}=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1375、 (3)、 设 $\displaystyle V$ 是全体 $\displaystyle 4$ 阶实矩阵关于矩阵的加法和数乘构成的实线性空间, \begin\{aligned\} V\_1=\left\\{A\in V; A^\mathrm\{T\} =A\right\\}, V\_2=\left\\{(a\_\{ij\})\in V; a\_\{ij\}=0, \forall\ i > j\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 \begin\{aligned\} \dim(V\_1+V\_2)-\dim(V\_1\cap V\_2)=\underline\{\ \ \ \ \ \ \ \ \ \ \}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (电子科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / \begin\{aligned\} \mbox\{原式\}=&\left\[\dim V\_1+\dim V\_2-\dim(V\_1\cap V\_2)\right\]-\dim(V\_1\cap V\_2)\\\\ =&\dim V\_1+\dim V\_2-2\dim(V\_1\cap V\_2)\\\\ =&(4+3+2+1)+(4+3+2+1)-2\cdot 4=12. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1376、 3、 设 $\displaystyle 4$ 阶实矩阵 $\displaystyle A$ 的秩为 $\displaystyle 2$, 线性无关的向量 $\displaystyle \alpha,\beta\in\mathbb\{R\}^4$ 使得 \begin\{aligned\} A(\alpha+\beta)=4\alpha+3\beta,\quad A(\alpha-\beta)=2\alpha-3\beta. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 试确定线性空间 $\displaystyle S=\left\\{B\in\mathbb\{R\}^\{4\times 4\}; AB=BA\right\\}$ 的维数. (电子科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 由题设, $\displaystyle A\alpha=3\alpha, A\beta=\alpha+3\beta$, 且 $\displaystyle \mathrm\{rank\} A=2\Rightarrow Ax=0$ 有两个线性无关的解 $\displaystyle \gamma,\delta$. 容易验证 $\displaystyle \alpha,\beta,\gamma,\delta$ 线性无关 [写出线性组合后用 $\displaystyle A$, $\displaystyle (A-3E),(A-3E)^2$ 作用即可]. 于是 \begin\{aligned\} A(\alpha,\beta,\gamma,\delta)=(\alpha,\beta,\gamma,\delta)J, J=\left(\begin\{array\}\{cccccccccccccccccccc\}3&1&&\\\\ &3&&\\\\ &&0&\\\\ &&&0\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 设 $\displaystyle P=(\alpha,\beta,\gamma,\delta)$, 则 $\displaystyle P$ 可逆, 而 \begin\{aligned\} B\in S\Leftrightarrow&P^\{-1\}BP\cdot P^\{-1\}AP=P^\{-1\}AP\cdot P^\{-1\}BP\\\\ \stackrel\{C=P^\{-1\}BP\}\{\Leftrightarrow\}& CJ=JC \Leftrightarrow C=\left(\begin\{array\}\{cccccccccccccccccccc\}a&b&&\\\\ &a&&\\\\ &&c&d\\\\ &&e&f\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle \dim S=7$, 且 $\displaystyle S$ 有一组基 \begin\{aligned\} &P^\{-1\} (E\_\{11\}+E\_\{22\}) P, P^\{-1\} E\_\{12\} P,\\\\ &P^\{-1\} E\_\{33\} P, P^\{-1\} E\_\{34\} P, P^\{-1\} E\_\{43\} P, P^\{-1\} E\_\{44\} P. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1377、 4、 设 $\displaystyle \mathbb\{F\}[x]\_4$ 为数域 $\displaystyle \mathbb\{F\}$ 上次数不超过 $\displaystyle 3$ 的多项式按加法和 $\displaystyle \mathbb\{F\}$-数乘构成的线性空间, 线性变换 \begin\{aligned\} \mathscr\{A\}: \mathbb\{F\}[x]\_4\to \mathbb\{F\}[x]\_4,\quad f(x)\mapsto f(2-x). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 试求 $\displaystyle \mathscr\{A\}$ 的特征多项式与最小多项式. (电子科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 由 \begin\{aligned\} &\mathscr\{A\} 1=1, \mathscr\{A\} x=2-x, \mathscr\{A\} x^2=4-4x+x^2,\\\\ &\mathscr\{A\} x^3=8-12x+6x^2-x^3 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 \begin\{aligned\} \mathscr\{A\} (1,x,x^2,x^3)=(1,x,x^2,x^3)A, A=\left(\begin\{array\}\{cccccccccccccccccccc\}1&2&4&8\\\\ 0&-1&-4&-12\\\\ 0&0&1&6\\\\ 0&0&0&-1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 易知 $\displaystyle A$ 的特征值为 $\displaystyle 1,1,-1,-1$. 又由 \begin\{aligned\} A-E\to \left(\begin\{array\}\{cccccccccccccccccccc\}0&1&2&0\\\\ 0&0&0&1\\\\ 0&0&0&0\\\\ 0&0&0&0\end\{array\}\right), A+E\to \left(\begin\{array\}\{cccccccccccccccccccc\}1&1&0&-2\\\\ 0&0&1&3\\\\ 0&0&0&0\\\\ 0&0&0&0\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle A$ 的属于特征值 $\displaystyle 1,-1$ 的特征向量各有两个. 故 $\displaystyle A\sim\mathrm\{diag\}(1,1,-1,-1)$. 故 $\displaystyle \mathscr\{A\}$ 的特征多项式与最小多项式分别为 \begin\{aligned\} (\lambda-1)^2(\lambda+1)^2,\quad (\lambda-1)(\lambda+1). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1378、 8、 证明题 (共 60 分, 每题 15 分). (1)、 设 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 是 $\displaystyle n$ 维复线性空间 $\displaystyle V$ 上的线性变换, 且 $\displaystyle \mathscr\{A\}\mathscr\{B\}=\mathscr\{B\}\mathscr\{A\}$, $\displaystyle \lambda$ 是 $\displaystyle \mathscr\{A\}$ 的一个负特征值, 此时 $\displaystyle \mathscr\{A\}$ 相应的特征子空间 \begin\{aligned\} V\_\lambda=\left\\{\alpha\in V; \mathscr\{A\}\alpha=\lambda \alpha\right\\}\neq \left\\{0\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (1-1)、 证明: $\displaystyle V\_\lambda$ 是一个 $\displaystyle \mathscr\{B\}$-子空间; (1-2)、 将 $\displaystyle \mathscr\{B\}$ 限制到 $\displaystyle V\_\lambda$ 上得到的线性变换记为 $\displaystyle \mathscr\{B\}|\_\{V\_\lambda\}$, 如果 $\displaystyle \mathscr\{B\}$ 可以对角化, 证明 $\displaystyle \mathscr\{B\}|\_\{V\_\lambda\}$ 也可以对角化; (1-3)、 若 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 都可以对角化, 证明: 存在 $\displaystyle V$ 的一组基 $\displaystyle \alpha\_1,\cdots,\alpha\_n$, 使得 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 在该基下的矩阵均为对角阵. (电子科技大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1-1)、 由 \begin\{aligned\} \alpha\in V\_\lambda\Rightarrow&\mathscr\{A\}\mathscr\{B\}\alpha=\mathscr\{B\}\mathscr\{A\}\alpha=\mathscr\{B\}(\lambda\alpha)=\lambda \mathscr\{B\}\alpha \Rightarrow \mathscr\{B\}\alpha\in V\_\lambda \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle V\_\lambda$ 是一个 $\displaystyle \mathscr\{B\}$-子空间. (1-2)、 记 $\displaystyle m\_\lambda(x)$, $\displaystyle m(x)$ 分别是 $\displaystyle \mathscr\{B\}|\_W$ 和 $\displaystyle \mathscr\{B\}$ 的最小多项式. 设 $\displaystyle \alpha\_1,\cdots,\alpha\_k$ 是 $\displaystyle W$ 的一组基, 将其扩充为 $\displaystyle V$ 的一组基 $\displaystyle \alpha\_1,\cdots,\alpha\_n$, 则 \begin\{aligned\} \mathscr\{B\}(\alpha\_1,\cdots,\alpha\_n)=(\alpha\_1,\cdots,\alpha\_n)\left(\begin\{array\}\{cc\} A&B\\\\ 0&C \end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 \begin\{aligned\} 0=m(\mathscr\{B\})(\alpha\_1,\cdots,\alpha\_n)=(\alpha\_1,\cdots,\alpha\_n)\left(\begin\{array\}\{cc\} m(A)&\star \\\\ 0&m(C) \end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 这样, $\displaystyle m(A)=0$, $\displaystyle m(x)$ 是 $\displaystyle \mathscr\{B\}|\_W$ 的零化多项式, 而 $\displaystyle m\_\lambda(x)\mid m(x)$. 既然 $\displaystyle \mathscr\{B\}$ 可对角化, 而 $\displaystyle m(x)$ 没有重根, $\displaystyle m\_\lambda(x)$ 更没有重根,$\mathscr\{B\}|\_\{V\_\lambda\}$ 可对角化. (1-3)、 由 $\displaystyle \mathscr\{A\}$ 可对角化知 \begin\{aligned\} V=\oplus\_\{i=1\}^s V\_\{\lambda\_i\}, V\_\{\lambda\_i\}=\left\\{\alpha\in V; \mathscr\{A\}\alpha=\lambda\_i\alpha\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 其中 $\displaystyle \lambda\_i$ 互异. 又由 $\displaystyle \mathscr\{B\}$ 可对角化及第 2 步知 $\displaystyle \mathscr\{B\}|\_\{V\_\{\lambda\_i\}\}$ 也可对角化. 设 $\displaystyle \varepsilon\_\{i1\},\cdots,\varepsilon\_\{in\_i\}$ 是 $\displaystyle V\_\{\lambda\_i\}$ 的基, 使得 \begin\{aligned\} &\mathscr\{B\}(\varepsilon\_\{i1\},\cdots,\varepsilon\_\{in\_i\})=(\varepsilon\_\{i1\},\cdots,\varepsilon\_\{in\_i\})\varLambda\_i,\\\\ &\varLambda\_i=\mathrm\{diag\}(\mu\_\{i1\},\cdots,\mu\_\{in\_i\}), n\_i=\dim V\_i. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 于是在 $\displaystyle V$ 的基 $\displaystyle \varepsilon\_\{11\},\cdots,\varepsilon\_\{1n\_1\}, \cdots, \varepsilon\_\{s1\},\cdots, \varepsilon\_\{sn\_s\}$ 下, $\displaystyle \mathscr\{A\}, \mathscr\{B\}$ 的矩阵分别为 \begin\{aligned\} &\mathrm\{diag\}(\lambda\_1E\_\{n\_1\},\cdots,\lambda\_sE\_\{n\_s\}),\\\\ &\mathrm\{diag\}(\varLambda\_1,\cdots,\varLambda\_s)=\mathrm\{diag\}(\mu\_\{11\},\cdots,\mu\_\{1n\_1\},\cdots,\mu\_\{s1\},\cdots,\mu\_\{sn\_s\}). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 结论得证.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1379、 4、 设 $\displaystyle \varepsilon\_1,\varepsilon\_2,\varepsilon\_3$ 是 $\displaystyle 3$ 维线性空间 $\displaystyle V$ 中的一组基. 已知 $\displaystyle V$ 上的线性变换 $\displaystyle \mathscr\{A\}$ 满足 \begin\{aligned\} \mathscr\{A\}\varepsilon\_1+2\mathscr\{A\} \varepsilon\_2+3\mathscr\{A\} \varepsilon\_3=&6\varepsilon\_1+7\varepsilon\_2+8\varepsilon\_3,\\\\ 3\mathscr\{A\} \varepsilon\_2+4\mathscr\{A\} \varepsilon\_3=&6\varepsilon\_1+9\varepsilon\_2+4\varepsilon\_3,\\\\ 4\mathscr\{A\} \varepsilon\_2+5\mathscr\{A\} \varepsilon\_3=&8\varepsilon\_1+12\varepsilon\_2+5\varepsilon\_3. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} (1)、 是否存在 $\displaystyle V$ 的一组基, 使得 $\displaystyle \mathscr\{A\}$ 在此其下的矩阵为对角阵, 是的话求对应的对角矩阵; 不是的话请说明理由. (2)、 求 $\displaystyle \mathscr\{A\}$ 在基 $\displaystyle \varepsilon\_1,\varepsilon\_1+\varepsilon\_2,\varepsilon\_1+\varepsilon\_2+\varepsilon\_3$ 下的矩阵. (东北大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 由题设, \begin\{aligned\} &\mathscr\{A\}(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ 2&3&4\\\\ 3&4&5\end\{array\}\right)=(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)\left(\begin\{array\}\{cccccccccccccccccccc\}6&6&8\\\\ 7&9&12\\\\ 8&4&5\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 \begin\{aligned\} &\mathscr\{A\}(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)=(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)A,\\\\ &A=\left(\begin\{array\}\{cccccccccccccccccccc\}6&6&8\\\\ 7&9&12\\\\ 8&4&5\end\{array\}\right)\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ 2&3&4\\\\ 3&4&5\end\{array\}\right)^\{-1\}=\left(\begin\{array\}\{cccccccccccccccccccc\}2&2&0\\\\ 1&3&0\\\\ 5&0&1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 易知 $\displaystyle A$ 的特征值为 $\displaystyle 4,1,1$. 又由 \begin\{aligned\} E-A=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ 0&1&0\\\\ 0&0&0\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle A$ 的 Jordan 标准形 $\displaystyle J$ 满足 $\displaystyle \mathrm\{rank\}(J-E)=\mathrm\{rank\}(A-E)=2$. 从而 $\displaystyle J$ 的严格上三角部分有一个 $\displaystyle 1$, $\displaystyle A$ (从而 $\displaystyle \mathscr\{A\}$) 不可对角化. (2)、 设 \begin\{aligned\} \eta\_1=\varepsilon\_1, \eta\_2=\varepsilon\_1+\varepsilon\_2, \eta\_3=\varepsilon\_1+\varepsilon\_2+\varepsilon\_3, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 \begin\{aligned\} (\eta\_1,\eta\_2,\eta\_3)=(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)T, T=\left(\begin\{array\}\{cccccccccccccccccccc\}1&1&1\\\\ 0&1&1\\\\ 0&0&1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 \begin\{aligned\} &\mathscr\{A\}(\eta\_1,\eta\_2,\eta\_3) =\mathscr\{A\}(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)T\\\\ =&(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)AT =(\eta\_1,\eta\_2,\eta\_3)T^\{-1\}AT. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 这表明 $\displaystyle \mathscr\{A\}$ 在基 $\displaystyle \eta\_1,\eta\_2,\eta\_3$ 下的矩阵为 \begin\{aligned\} T^\{-1\}AT=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ -4&-1&-2\\\\ 5&5&6\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} \begin\{aligned\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/ --- 1380、 7、 设 $\displaystyle V\_1$ 是数域 $\displaystyle \mathbb\{P\}$ 上 $\displaystyle n$ 阶对称矩阵全体构成的集合, $\displaystyle V\_2$ 是数域 $\displaystyle \mathbb\{P\}$ 上 $\displaystyle n$ 阶反对称矩阵全体构成的集合. (1)、 证明: $\displaystyle V\_1$ 与 $\displaystyle V\_2$ 都是 $\displaystyle \mathbb\{P\}^\{n\times n\}$ 的线性子空间; (2)、 证明: 任一 $\displaystyle n$ 阶矩阵可分解为一个对称矩阵与一个反对称矩阵的和; (3)、 证明: $\displaystyle \mathbb\{P\}^\{n\times n\}$ 可表示为 $\displaystyle V\_1$ 与 $\displaystyle V\_2$ 的直和, 即 $\displaystyle \mathbb\{P\}^\{n\times n\}=V\_1\oplus V\_2$. (东北大学2023年高等代数考研试题) [线性空间与线性变换 ] [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (1)、 对 $\displaystyle \forall\ k,l\in\mathbb\{P\}$, \begin\{aligned\} &A,B\in V\_1\Rightarrow A^\mathrm\{T\} =A, B^\mathrm\{T\} =B\\\\ \Rightarrow& (kA+lB)^\mathrm\{T\}=kA^\mathrm\{T\}+lB^\mathrm\{T\} =kA+lB\Rightarrow kA+lB\in V\_1,\\\\ &A,B\in V\_2\Rightarrow A^\mathrm\{T\} =-A, B^\mathrm\{T\} =-B\\\\ \Rightarrow& (kA+lB)^\mathrm\{T\}=kA^\mathrm\{T\}+lB^\mathrm\{T\} =-kA-lB\Rightarrow kA+lB\in V\_2. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 故 $\displaystyle V\_1$ 与 $\displaystyle V\_2$ 都是 $\displaystyle \mathbb\{P\}^\{n\times n\}$ 的线性子空间. (2)、 对 $\displaystyle \forall\ A\in\mathbb\{P\}^\{n\times n\}$, \begin\{aligned\} A=\frac\{A+A^\mathrm\{T\}\}\{2\}+\frac\{A-A^\mathrm\{T\}\}\{2\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 是一个对称矩阵与一个反对称矩阵的和. (3)、 由第 2 步知 $\displaystyle \mathbb\{P\}^\{n\times n\}=V\_1+V\_2$. 又由 \begin\{aligned\} A\in V\_1\cap V\_2\Rightarrow A^\mathrm\{T\} =A, A^\mathrm\{T\}=-A\Rightarrow A=0 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle \mathbb\{P\}^\{n\times n\}=V\_1\oplus V\_2$.跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/
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