## 张祖锦2023年数学专业真题分类70天之第66天
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1496、 (5)、 若 $\displaystyle W$ 是实线性空间 $\displaystyle \mathbb\{R\}^5$ 的非零子空间, 且 $\displaystyle W$ 中任意非零向量的分量都不为零, 则 $\displaystyle \dim W=\underline\{\ \ \ \ \ \ \ \ \ \ \}$. (云南大学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 $\displaystyle W$ 是非零子空间知 $\displaystyle \dim W\geq 1$. 往用反证法证明 $\displaystyle \dim W=1$. 若不然, $\displaystyle \dim W\geq 2$, 则 $\displaystyle W$ 有线性无关的向量 $\displaystyle \alpha=\left(\begin\{array\}\{cccccccccccccccccccc\}a\_1\\\\\vdots\\\\a\_5\end\{array\}\right), \beta=\left(\begin\{array\}\{cccccccccccccccccccc\}b\_1\\\\\vdots\\\\b\_5\end\{array\}\right)$, 则
\begin\{aligned\} \alpha\neq 0, \beta\neq 0, k\alpha+l\beta=0\Rightarrow k=l=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
不妨设 $\displaystyle a\_1\neq 0$, 则
\begin\{aligned\} 1\neq 0\Rightarrow 0\neq -\frac\{b\_1\}\{a\_1\}\alpha+1\cdot \beta\in W, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
但 $\displaystyle 0\neq -\frac\{b\_1\}\{a\_1\}\alpha+1\cdot \beta$ 的第一个分量为 $\displaystyle -\frac\{b\_1\}\{a\_1\}\cdot a\_1+1\cdot b\_1=0$. 这与 $\displaystyle W$ 的定义矛盾. 故 $\displaystyle \dim W=1$.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)/
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1497、 7、 (15 分) 设 $\displaystyle \sigma,\tau$ 是非零的有限维复线性空间 $\displaystyle V$ 上的任意两个线性变换. 证明: $\displaystyle \sigma\tau-\tau\sigma$ 不可能是 $\displaystyle V$ 上的恒等变换. 举例说明若 $\displaystyle V$ 是无限维线性空间, 则此结论不成立. (云南大学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)、 当 $\displaystyle \dim V=n < \infty$ 时, 用反证法证明. 若 $\displaystyle \sigma\tau-\tau\sigma=\mathscr\{E\}$, 则设 $\displaystyle \sigma,\tau$ 在 $\displaystyle V$ 的基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$ 下的矩阵分别为 $\displaystyle A,B$ 后,
\begin\{aligned\} AB-BA=E\Rightarrow 0=\mathrm\{tr\}(AB-BA)=\mathrm\{tr\} E=n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
这是一个矛盾. 故有结论. (2)、 当 $\displaystyle V$ 是无限维时, $\displaystyle \sigma\tau-\tau\sigma$ 可能为恒等变换. 比如 $\displaystyle \mathbb\{P\}[x]$ 是 $\displaystyle \mathbb\{R\}$ 上全体多项式按通常的加法和数于多项式的乘法构成的线性空间,
\begin\{aligned\} &V=\left\\{f(x)\in\mathbb\{P\}[x]; f(0)=0\right\\},\\\\ &\sigma\left\[f(x)\right\]=f'(x), \tau\left\[f(x)\right\]=xf(x), \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\} &\sigma\tau \left\[f(x)\right\]-\tau\sigma\left\[f(x)\right\] =\sigma\left\[xf(x)\right\]-\tau\left\[f'(x)\right\]\\\\ =&[xf(x)]'-xf'(x)=f(x)=\mathscr\{E\} \left\[f(x)\right\]. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
故 $\displaystyle \sigma\tau-\tau\sigma=\mathscr\{E\}$.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](
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1498、 (0-4)、 已知向量 $\displaystyle \alpha\_1=\left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\0\\\\3\end\{array\}\right), \alpha\_2=\left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\-1\\\\a\end\{array\}\right), \alpha\_3=\left(\begin\{array\}\{cccccccccccccccccccc\}2\\\\a+1\\\\1\end\{array\}\right), \beta=\left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\1\\\\b+2\end\{array\}\right)$, 当 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$ 时, $\displaystyle \beta$ 可被 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3$ 线性表出且表示法不唯一. (长安大学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\} &(\alpha\_1,\alpha\_2,\alpha\_3,\beta) =\left(\begin\{array\}\{cccccccccccccccccccc\}1&1&2&1\\\\ 0&-1&a+1&1\\\\ 3&a&1&b+2\end\{array\}\right)\\\\ \to& \left(\begin\{array\}\{cccccccccccccccccccc\}1&1&2&1\\\\ 0&1&-a-1&-1\\\\ 0&a-3&-5&b-1\end\{array\}\right)\\\\ \to&\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&a+3&2\\\\ 0&1&-a-1&-1\\\\ 0&0&(a-4)(a+2)&a+b-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\}
知
\begin\{aligned\} (a-4)(a+2)=0, a+b-4=0\Rightarrow a=4, b=0\mbox\{或\} a=-2, b=6. \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)/
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1499、 7、 (10 分) 设 $\displaystyle \mathscr\{A\}$ 是 $\displaystyle n$ 维线性空间 $\displaystyle V$ 上的线性变换, 证明: $\displaystyle \mathscr\{A\}$ 的秩 $\displaystyle +\mathscr\{A\}$ 的零度 $\displaystyle =n$. (长安大学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 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_r$ 是 $\displaystyle \ker \mathscr\{A\}$ 的一组基, 将其扩充为 $\displaystyle V$ 的一组基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$, 则由
\begin\{aligned\} &\sum\_\{i=r+1\}^n k\_i\mathscr\{A\}\varepsilon\_i=0\Rightarrow \mathscr\{A\}\left(\sum\_\{i=r+1\}^n k\_i\varepsilon\_i\right)=0 \Rightarrow \sum\_\{i=r+1\}^n k\_i\varepsilon\_i\in \ker \mathscr\{A\}\\\\ \Rightarrow& \exists\ 1\leq i\leq r,\mathrm\{ s.t.\} \sum\_\{i=r+1\}^n k\_i\varepsilon\_i =-\sum\_\{i=1\}^r k\_i\varepsilon\_i \Rightarrow k\_i=0 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle \mathscr\{A\}\varepsilon\_\{r+1\},\cdots,\mathscr\{A\}\varepsilon\_n$ 线性无关, 而是 $\displaystyle \mathrm\{im\} \mathscr\{A\}$ 的一组基. 于是$\mathscr\{A\}$ 的零度 $\displaystyle +\mathscr\{A\}$ 的秩
\begin\{aligned\} =\dim \ker \mathscr\{A\}+\dim \mathrm\{im\} \mathscr\{A\}=r+(n-r)=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)/
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1500、 (3)、 设 $\displaystyle \mathbb\{F\}$ 是一个数域,
\begin\{aligned\} W=\left\\{A\in\mathbb\{F\}^\{n\times n\}; A^\mathrm\{T\} =A\right\\}, V=\left\\{A\in\mathbb\{F\}^\{n\times n\}; A^\mathrm\{T\}=-A\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
则 $\displaystyle \dim(W+V)=\underline\{\ \ \ \ \ \ \ \ \ \ \}$. (郑州大学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) / $\displaystyle \dim(W+V)=\dim\mathbb\{F\}^\{n\times n\}=n^2$.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)/
---
1501、 (2)、 设
\begin\{aligned\} &\alpha\_1=(1+a,1,1,1)^\mathrm\{T\}, \alpha\_2=(2,2+a,2,2)^\mathrm\{T\},\\\\ &\alpha\_3=(3,3,3+a,3)^\mathrm\{T\}, \alpha\_4=(4,4,4,4+a)^\mathrm\{T\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
(2-1)、 当 $\displaystyle a$ 满足什么条件时, $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4$ 线性相关. (2-2)、 在 (1) 的情形下, 求 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4$ 的极大线性无关组, 并将剩余向量用极大线性无关组线性表出. (郑州大学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)、 由
\begin\{aligned\} \det(\alpha\_1,\cdots,\alpha\_4)=a^3(a+10) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知当且仅当 $\displaystyle a=0\mbox\{或\} a=-10$ 时, $\displaystyle \det A=0$, $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4$ 线性相关. (2-2)、 若 $\displaystyle a=0$, 则 $\displaystyle \alpha\_1$ 就是一个极大线性无关组, 且 $\displaystyle \alpha\_2=\alpha\_1,\alpha\_3=\alpha\_1, \alpha\_4=\alpha\_1$. 若 $\displaystyle a=-10$, 则由
\begin\{aligned\} (\alpha\_1,\cdots,\alpha\_4)\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&-1\\\\ 0&1&0&-1\\\\ 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\}
知 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3$ 是一个极大无关组, 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\_4=-\alpha\_1-\alpha\_2-\alpha\_3. \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)/
---
1502、 3、 已知 $\displaystyle V$ 是数域 $\displaystyle \mathbb\{P\}$ 上的 $\displaystyle n$ 维线性空间, $\displaystyle W\_1,W\_2$ 是 $\displaystyle V$ 的子空间, 且满足
\begin\{aligned\} \dim W\_1+\dim W\_2=n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
证明: 存在 $\displaystyle V$ 上的线性变换 $\displaystyle \sigma$, 使得 $\displaystyle \ker \sigma=W\_1, \mathrm\{im\} \sigma=W\_2$. (郑州大学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 $\displaystyle \dim W\_1=r, \dim W\_2=n-r$. 取定 $\displaystyle W\_1$ 的一组基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_r$, 并将其扩充为 $\displaystyle V$ 的一组基 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$. 再取 $\displaystyle W\_2$ 的一组基 $\displaystyle \eta\_\{r+1\},\cdots,\eta\_n$. 作 $\displaystyle V$ 上的线性变换 $\displaystyle \sigma$ 使得
\begin\{aligned\} \sigma(\varepsilon\_i)=\left\\{\begin\{array\}\{llllllllllll\}0,&1\leq i\leq r,\\\\ \eta\_i,&r+1\leq i\leq n.\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\} \mathrm\{im\} \sigma&=L\left(\sigma(\varepsilon\_1),\cdots,\sigma(\varepsilon\_n)\right) =L\left(\sigma(\varepsilon\_\{r+1\}),\cdots,\sigma(\varepsilon\_n)\right)\\\\ &=L(\eta\_\{r+1\},\cdots,\eta\_n)=W\_2. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
又由
\begin\{aligned\} &\alpha=\sum\_\{i=1\}^n k\_i\varepsilon\_i\in \ker \sigma \Leftrightarrow 0=\sigma(\alpha)=\sum\_\{i=r+1\}^n k\_i\eta\_i\\\\ \Leftrightarrow& k\_i=0, r+1\leq i\leq n \Leftrightarrow \alpha=\sum\_\{i=1\}^r k\_i\varepsilon\_i \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle \ker \sigma=L(\varepsilon\_1,\cdots,\varepsilon\_r)=W\_1$.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)/
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1503、 4、 设 $\displaystyle f(x),g(x)$ 为数域 $\displaystyle \mathbb\{P\}$ 上的互素多项式, $\displaystyle A\in\mathbb\{P\}^\{n\times n\}$. 证明:
\begin\{aligned\} \ker\left(f(A)g(A)\right)=\ker f(A)\oplus \ker g(A). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
(郑州大学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) / 易知 $\displaystyle \ker f(A)\subset \ker\left(f(A)g(A)\right), \ker g(A)\subset \ker\left(f(A)g(A)\right)$. 由
\begin\{aligned\} (f,g)=1\Rightarrow& \exists\ u,v,\mathrm\{ s.t.\} uf+vg=1\\\\ \Rightarrow&E\_n=u(A)f(A)+v(A)g(A). \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\} \alpha\in \mathbb\{P\}^n\Rightarrow&\alpha=E\_n\alpha=v(A)g(A)\alpha+u(A)f(A)\alpha\\\\ &\in \ker f(A)+\ker g(A),\\\\ \alpha\in \ker f(A)\cap \ker g(A)\Rightarrow&f(A)\alpha=0, g(A)\alpha=0\\\\ \Rightarrow&\alpha=E\_n\alpha=v(A)g(A)\alpha+u(A)f(A)\alpha=0. \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 $\displaystyle \ker\left(f(A)g(A)\right)=\ker f(A)\oplus \ker g(A)$.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](
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---
1504、 (3)、 (20 分) 考虑二阶复方阵 $\displaystyle M\_2(\mathbb\{C\})$ 组成的复线性空间, 方阵 $\displaystyle A=\left(\begin\{array\}\{cccccccccccccccccccc\}8&1\\\\ 3&8\end\{array\}\right)$. 定义线性变换
\begin\{aligned\} \mathscr\{B\}: M\_2(\mathbb\{C\})\to M\_2(\mathbb\{C\}) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
使得 $\displaystyle \mathscr\{B\}(X)=AX-XA$, 其中 $\displaystyle X$ 为任意二阶复方阵. 试求线性变换 $\displaystyle \mathscr\{B\}$ 的特征值, 相应的特征子空间以及最小多项式. (中国科学技术大学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\{B\} E\_\{11\}=\left(\begin\{array\}\{cccccccccccccccccccc\}0&-1\\\\3&0\end\{array\}\right), \mathscr\{B\} E\_\{12\}=\left(\begin\{array\}\{cccccccccccccccccccc\}-3&0\\\\0&3\end\{array\}\right),\\\\ \mathscr\{B\} E\_\{21\}=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0\\\\0&-1\end\{array\}\right), \mathscr\{B\} E\_\{22\}=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\-3&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\} \mathscr\{B\}(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\}) =&(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\})B,\\\\ B=&\left(\begin\{array\}\{cccccccccccccccccccc\} 0&-3&1&0\\\\ -1&0&0&1\\\\ 3&0&0&-3\\\\ 0&3&-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\}
易知 $\displaystyle B$ 的特征值为 $\displaystyle 2\sqrt\{3\}, -2\sqrt\{3\},0,0$. 由
\begin\{aligned\} 2\sqrt\{3\}E\_4-B\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&1\\\\ 0&1&0&-\frac\{1\}\{\sqrt\{3\}\}\\\\ 0&0&1&\sqrt\{3\}\\\\ 0&0&0&0\end\{array\}\right),\\\\ -2\sqrt\{3\}E\_4-B\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&1\\\\ 0&1&0&\frac\{1\}\{\sqrt\{3\}\}\\\\ 0&0&1&-\sqrt\{3\}\\\\ 0&0&0&0\end\{array\}\right),\\\\ 0E\_4-B\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&-1\\\\ 0&1&-\frac\{1\}\{3\}&0\\\\ 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 $\displaystyle B$ 的属于特征值 $\displaystyle 2\sqrt\{3\}, -2\sqrt\{3\},0$ 的特征向量为
\begin\{aligned\} \xi\_1=\left(\begin\{array\}\{cccccccccccccccccccc\}-\sqrt\{3\}\\\\1\\\\-3\\\\\sqrt\{3\}\end\{array\}\right); \quad \xi\_2=\left(\begin\{array\}\{cccccccccccccccccccc\}-\sqrt\{3\}\\\\-1\\\\3\\\\\sqrt\{3\}\end\{array\}\right);\quad \xi\_3=\left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\1\\\\3\\\\0\end\{array\}\right), \xi\_4=\left(\begin\{array\}\{cccccccccccccccccccc\}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\}
故线性变换 $\displaystyle \mathscr\{B\}$ 的特征子空间为
\begin\{aligned\} V\_\{2\sqrt\{3\}\}=L\left(\left(\begin\{array\}\{cccccccccccccccccccc\}-\sqrt\{3\}&1\\\\-3&\sqrt\{3\}\end\{array\}\right)\right), V\_\{-2\sqrt\{3\}\}=L\left(\left(\begin\{array\}\{cccccccccccccccccccc\}-\sqrt\{3\}&-1\\\\3&\sqrt\{3\}\end\{array\}\right)\right),\\\\ V\_0=L\left(\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\3&0\end\{array\}\right), E\_4\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
由 $\displaystyle B$ 有 $\displaystyle 4$ 个线性无关的特征向量知 $\displaystyle B$ 可对角化, 而 $\displaystyle \mathscr\{B\}$ 的最小多项式为
\begin\{aligned\} (\lambda-2\sqrt\{3\})(\lambda+2\sqrt\{3\})\lambda=\lambda^3-12\lambda. \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)/
---
1505、 5、 设 $\displaystyle M\_\{2\times 2\}$ 是二阶矩阵集合,
\begin\{aligned\} A=\left(\begin\{array\}\{cccccccccccccccccccc\}1&-1\\\\ 2&0\end\{array\}\right),\quad B=\left(\begin\{array\}\{cccccccccccccccccccc\}4&0\\\\ 3&1\end\{array\}\right),\quad L: X\mapsto AXB. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
求 $\displaystyle L$ 的迹和行列式. (中国科学院大学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\} L E\_\{11\}=\left(\begin\{array\}\{cccccccccccccccccccc\}4&0\\\\8&0\end\{array\}\right), L E\_\{12\}=\left(\begin\{array\}\{cccccccccccccccccccc\}3&1\\\\6&2\end\{array\}\right),\\\\ L E\_\{21\}=\left(\begin\{array\}\{cccccccccccccccccccc\}-4&0\\\\0&0\end\{array\}\right), L E\_\{22\}=\left(\begin\{array\}\{cccccccccccccccccccc\}-3&-1\\\\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
\begin\{aligned\} L (E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\}) =&(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\})C,\\\\ C=&\left(\begin\{array\}\{cccccccccccccccccccc\} 4&3&-4&-3\\\\ 0&1&0&-1\\\\ 8&6&0&0\\\\ 0&2&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 $\displaystyle L$ 的迹为 $\displaystyle \mathrm\{tr\} C=5$, $\displaystyle L$ 的行列式为 $\displaystyle \det C=64$.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](
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---
1506、 6、 设 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 是 $\displaystyle \mathbb\{C\}^n$ 上的线性变换, 且 $\displaystyle \mathscr\{A\}^2=\mathscr\{B\}^2=\mathscr\{E\}$, 其中 $\displaystyle \mathscr\{E\}$ 为恒等变换. 证明: (1)、 若 $\displaystyle n$ 为奇数, 则 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 有公共特征向量; (2)、 若 $\displaystyle n$ 为偶数, 则存在一个子空间 $\displaystyle W$, 同时为 $\displaystyle \mathscr\{A\},\mathscr\{B\}$ 的不变子空间, 且 $\displaystyle \dim W=1\mbox\{或\} 2$. (中国科学院大学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)、 设
\begin\{aligned\} V\_\{\pm 1\}=\left\\{\alpha\in\mathbb\{C\}^n; \mathscr\{A\}\alpha=\pm \alpha\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
则由
\begin\{aligned\} &\mathbb\{C\}^n\ni \alpha=\frac\{\alpha+\mathscr\{A\}\alpha\}\{2\}+\frac\{\alpha-\mathscr\{A\}\alpha\}\{2\}\in V\_1+V\_\{-1\},\\\\ &\alpha\in V\_1\cap V\_\{-1\}\Rightarrow \alpha=\mathscr\{A\}\alpha=-\alpha\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\}
知 $\displaystyle \mathbb\{C\}^n=V\_1\oplus V\_\{-1\}$, $\displaystyle n=\dim V\_1+\dim V\_\{-1\}$. 再设
\begin\{aligned\} W\_\{\pm 1\}=\left\\{\alpha\in\mathbb\{C\}^n; \mathscr\{B\}\alpha=\alpha\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
则也有 $\displaystyle n=\dim W\_1+\dim W\_\{-1\}$. 于是
\begin\{aligned\} &\exists\ i\in \left\\{-1,1\right\\},\mathrm\{ s.t.\} \dim V\_i\geq \frac\{n\}\{2\}\Rightarrow \dim V\_i\geq \frac\{n+1\}\{2\},\\\\ &\exists\ j\in \left\\{-1,1\right\\},\mathrm\{ s.t.\} \dim W\_j\geq \frac\{n\}\{2\}\Rightarrow \dim W\_j\geq \frac\{n+1\}\{2\}. \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\} \dim(V\_i\cap W\_j)=&\dim V\_i+\dim W\_j-\dim(V\_i+W\_j)\\\\ \geq&\frac\{n+1\}\{2\}+\frac\{n+1\}\{2\}-n=1. \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\} \exists\ \alpha\in V\_i\cap V\_j\Rightarrow \mathscr\{A\}\alpha=i\alpha, \mathscr\{B\}\alpha=j\alpha. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
此 $\displaystyle \alpha$ 就是 $\displaystyle \mathscr\{A\}, \mathscr\{B\}$ 的公共特征向量. (2)、 考虑 $\displaystyle \mathscr\{A\}\mathscr\{B\}: \mathbb\{C\}^n\to\mathbb\{C\}^n$, 它有一个复特征值 $\displaystyle \lambda$. 设对应的特征向量为 $\displaystyle \alpha\neq 0$, 则 $\displaystyle \mathscr\{A\}\mathscr\{B\}\alpha=\lambda \alpha$. 考虑 $\displaystyle W=L(\alpha,\mathscr\{A\}\alpha)$, 则 $\displaystyle W$ 自然是 $\displaystyle \mathscr\{A\}$ 不变的. 又由
\begin\{aligned\} &\mathscr\{B\}\alpha=\mathscr\{A\}\mathscr\{A\}\mathscr\{B\}\alpha =\mathscr\{A\}(\lambda \alpha)=\lambda \mathscr\{A\}\alpha,\qquad(I)\\\\ &\lambda \mathscr\{B\}\mathscr\{A\}\alpha=\mathscr\{B\}\mathscr\{A\}(\lambda \alpha) =\mathscr\{B\}(\lambda \mathscr\{A\}\alpha)\stackrel\{(I)\}\{=\}\mathscr\{B\}\mathscr\{B\}\alpha=\alpha\neq 0\\\\ \Rightarrow&\lambda\neq 0, \mathscr\{B\}\mathscr\{A\}\alpha=\frac\{1\}\{\lambda\}\alpha \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle W$ 也是 $\displaystyle \mathscr\{B\}$ 不变的. 由 $\displaystyle \alpha\neq 0$ 蕴含 $\displaystyle \dim W=1\mbox\{或\} 2$.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](
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---
1507、 8、 设 $\displaystyle U,V,W$ 是某一线性空间的子空间, 证明:
\begin\{aligned\} (U+V)\cap (U+W)\cap (W+V)=U\cap (W+V)+V\cap(U+W). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
[题目有问题, 毕竟 $\displaystyle U\cap (W+V)\subset U+V$ 一般不成立啊. 所以张祖锦没法做哦.] (中国科学院大学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, 毕竟 $\displaystyle U\cap (W+V)\subset U+V$ 一般不成立啊. 所以张祖锦没法做哦.]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)/
---
1508、 (3)、 $\displaystyle m$ 个 $\displaystyle n$ 维向量空间的一个部分向量组是线性相关的, 则这个向量组是 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$.
A. 线性相关
B. 线性无关
C. 可能线性相关, 也可能线性无关
D. 以上都不对 (中国矿业大学(北京)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) / $\displaystyle A$.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)/
---
1509、 (7)、 $\displaystyle V$ 是 $\displaystyle n$ 维线性空间, $\displaystyle \mathrm\{Hom\}(V)$ 表示 $\displaystyle V$ 上所有线性变换的集合, 它也是线性空间, 则 $\displaystyle \mathrm\{Hom\}(V)$ 的维数是 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$.
A. $\displaystyle n^2$
B. $\displaystyle n$
C. $\displaystyle 2n$
D. $\displaystyle \infty$ (中国矿业大学(北京)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) / $\displaystyle A$.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)/
---
1510、 9、 设 $\displaystyle V$ 是 $\displaystyle n$ 维线性空间, $\displaystyle \mathscr\{A\}$ 是其上的一个线性变换, 它在 $\displaystyle V$ 的一组基下的矩阵为 $\displaystyle A$, 定义 $\displaystyle \mathrm\{rank\} \mathscr\{A\}=\mathrm\{rank\} A$. 证明: (1)、 $\displaystyle \mathrm\{rank\} \mathscr\{A\}$ 不依赖于基的选取; (2)、 若 $\displaystyle \mathrm\{rank\} \mathscr\{A\}=\mathrm\{rank\}(\mathscr\{A\}^2)$, 则 $\displaystyle \mathscr\{A\} V+\mathscr\{A\}^\{-1\}(0)$ 是直和. (中国矿业大学(北京)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)、 设 $\displaystyle \varepsilon\_1,\cdots,\varepsilon\_n$ 与 $\displaystyle \eta\_1,\cdots,\eta\_n$ 是 $\displaystyle V$ 的两组基. 再设
\begin\{aligned\} \mathscr\{A\}(\varepsilon\_1,\cdots,\varepsilon\_n)=&(\varepsilon\_1,\cdots,\varepsilon\_n)A,\\\\ \mathscr\{A\}(\eta\_1,\cdots,\eta\_n)=&(\eta\_1,\cdots,\eta\_n)B,\\\\ (\eta\_1,\cdots,\eta\_n)=&(\varepsilon\_1,\cdots,\varepsilon\_n)T, |T|\neq 0, \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\} &\mathscr\{A\}(\eta\_1,\cdots,\eta\_n)=\mathscr\{A\}(\varepsilon\_1,\cdots,\varepsilon\_n)T\\\\ =&(\varepsilon\_1,\cdots,\varepsilon\_n)AT =(\eta\_1,\cdots,\eta\_n)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\}
于是 $\displaystyle B=T^\{-1\}AT\Rightarrow \mathrm\{rank\} B=\mathrm\{rank\} A$. 从而 $\displaystyle \mathrm\{rank\} \mathscr\{A\}$ 不依赖于基的选取. (2)、 易知 $\displaystyle \ker \mathscr\{A\}\subset \ker (\mathscr\{A\}^2)$. 又由
\begin\{aligned\} \dim \ker \mathscr\{A\}=n-\mathrm\{rank\} \mathscr\{A\}=n-\mathrm\{rank\}(\mathscr\{A\}^2)=\dim \ker (\mathscr\{A\}^2) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle \ker \mathscr\{A\}=\ker (\mathscr\{A\}^2)$. 于是
\begin\{aligned\} &\alpha\in \ker\mathscr\{A\}\cap \mathrm\{im\} \mathscr\{A\}\Rightarrow \mathscr\{A\}\alpha=0; \exists\ \beta\in V,\mathrm\{ s.t.\} \mathscr\{A\}\beta=\alpha\\\\ \Rightarrow&0=\mathscr\{A\}\alpha=\mathscr\{A\}^2\beta \Rightarrow \beta\in \ker (\mathscr\{A\}^2)=\ker \mathscr\{A\}\Rightarrow \alpha=\mathscr\{A\}\beta=0. \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 $\displaystyle \ker \mathscr\{A\}\cap \mathrm\{im\} \mathscr\{A\}=\left\\{0\right\\}$,
\begin\{aligned\} \dim\left(\ker \mathscr\{A\}+\mathrm\{im\} \mathscr\{A\}\right)=&\dim \ker \mathscr\{A\}+\dim \mathrm\{im\} \mathscr\{A\}-\dim\left(\ker \mathscr\{A\}\cap \mathrm\{im\} \mathscr\{A\}\right)\\\\ =&n-0=n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
于是 $\displaystyle \ker \mathscr\{A\}+\mathrm\{im\} \mathscr\{A\}=V$. 联合 $\displaystyle \ker \mathscr\{A\}\cap \mathrm\{im\} \mathscr\{A\}=\left\\{0\right\\}$ 知 $\displaystyle V=\ker \mathscr\{A\}\oplus\mathrm\{im\} \mathscr\{A\}$.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)/
---
1511、 (6)、 记全体实数为 $\displaystyle \mathbb\{R\}$, 已知实矩阵
\begin\{aligned\} &A\_1=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ -1&-1\end\{array\}\right), A\_2=\left(\begin\{array\}\{cccccccccccccccccccc\}0&3\\\\ 0&-1\end\{array\}\right), A\_3=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0\\\\ 1&3\end\{array\}\right),\\\\ &A\_4=\left(\begin\{array\}\{cccccccccccccccccccc\}0&2\\\\ 1&-2\end\{array\}\right), A\_5=\left(\begin\{array\}\{cccccccccccccccccccc\}0&5\\\\ -2&-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\}
则 $\displaystyle \mathbb\{R\}^\{4\times 4\}$ 的生成子空间 $\displaystyle L(A\_1,A\_2,A\_3,A\_4,A\_5)$ 的维数是 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (中国矿业大学(徐州)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\_1,\cdots,A\_5)=(E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\})A,\\\\ &A=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0&0&0&0\\\\ 1&3&0&2&5\\\\ -1&0&1&1&-2\\\\ -1&-1&3&-2&-6\end\{array\}\right)\to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&-\frac\{13\}\{7\}&\frac\{5\}\{7\}\\\\ 0&1&0&\frac\{9\}\{7\}&\frac\{10\}\{7\}\\\\ 0&0&1&-\frac\{6\}\{7\}&-\frac\{9\}\{7\}\\\\ 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 $\displaystyle \mathrm\{rank\} A=3$, 而 $\displaystyle \dim L(A\_1,A\_2,A\_3,A\_4,A\_5)=\mathrm\{rank\} A=3$.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)/
---
1512、 (8)、 设 $\displaystyle V$ 是数域 $\displaystyle \mathbb\{P\}$ 上全体 $\displaystyle 2$ 阶矩阵构成的线性空间, 对于给定的 $\displaystyle A\in V$, 定义 $\displaystyle V$ 上的线性变换 $\displaystyle \sigma$:
\begin\{aligned\} \sigma(B)=AB-BA, \forall\ B\in V. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
如果 $\displaystyle A$ 是幂零的, 则 $\displaystyle \sigma$ 在基 $\displaystyle E\_\{11\},E\_\{12\},E\_\{21\},E\_\{22\}$ ($E\_\{ij\})$ 表示 $\displaystyle (i,j)$ 元为 $\displaystyle 1$, 其余元素为 $\displaystyle 0$ 的矩阵) 下的矩阵 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$ 幂零的. (选题: 一定是, 未必是) (中国矿业大学(徐州)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) / 把 $\displaystyle A$ 看成复矩阵, 则 $\displaystyle A$ 的特征值全为 $\displaystyle 0$, 而存在复可逆矩阵 $\displaystyle P$ 使得
\begin\{aligned\} P^\{-1\}AP=0\mbox\{或\} \left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ 0&0\end\{array\}\right)\equiv J. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
若 $\displaystyle J=0$, 则 $\displaystyle A=0, \sigma(B)=0$, $\displaystyle \sigma$ 在基 $\displaystyle E\_\{11\},\cdots,E\_\{22\}$ 下的矩阵 $\displaystyle C=0$, 自然是幂零矩阵. 若 $\displaystyle J=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ 0&0\end\{array\}\right)$, 则
\begin\{aligned\} P^\{-1\}\sigma(E\_\{ij\})P=P^\{-1\}(AE\_\{ij\}-E\_\{ij\}A)P =J\tilde\{E\}\_\{ij\}-\tilde\{E\}\_\{ij\}J, \qquad(I) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
其中 $\displaystyle \tilde\{E\}\_\{ij\}=P^\{-1\}E\_\{ij\}P$. 设
\begin\{aligned\} &\sigma(E\_\{11\},\cdots,E\_\{22\})=(E\_\{11\},\cdots,E\_\{22\})C,\\\\ &\sigma\_J(\tilde\{E\}\_\{11\},\cdots,\tilde\{E\}\_\{22\})=((\tilde\{E\}\_\{11\},\cdots,\tilde\{E\}\_\{22\})\tilde\{C\},\\\\ &\varepsilon\_1=E\_\{11\},\cdots, \varepsilon\_4=E\_\{22\}; \tilde\{\varepsilon\}\_1=\tilde\{E\}\_\{11\}, \cdots, \tilde\{\varepsilon\}\_4=\tilde\{E\}\_\{22\}, \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\} \sigma(\varepsilon\_i)=\sum\_k c\_\{ki\}\varepsilon\_k, \sigma(\tilde\{\varepsilon\}\_i)=\sum\_k\tilde\{c\}\_\{ki\}\tilde\{E\}\_k. \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\} P^\{-1\}\sigma(\varepsilon\_i)P=&P^\{-1\}\left(\sum\_k c\_\{ki\}\varepsilon\_k\right)P =\sum\_k c\_\{ki\}P^\{-1\}\varepsilon\_kP =\sum\_k c\_\{ki\}\tilde\{\varepsilon\}\_k,\\\\ \sigma\_j\left(\tilde\{\varepsilon\}\_i\right)=&\sum\_k \tilde\{c\}\_\{ki\}\tilde\{\varepsilon\}\_k. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
代入 $\displaystyle (I)$ 知
\begin\{aligned\} \tilde\{c\}\_\{ki\}=c\_\{ki\}\Leftrightarrow \tilde\{C\}=C. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
由
\begin\{aligned\} &\sigma\_J(E\_\{11\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&-1\\\\ 0&0\end\{array\}\right), \sigma\_J(E\_\{12\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0\\\\ 0&\end\{array\}\right),\\\\ &\sigma\_J(E\_\{21\})=\left(\begin\{array\}\{cccccccccccccccccccc\}1&0\\\\ 0&-1\end\{array\}\right), \sigma\_J(E\_\{22\})=\left(\begin\{array\}\{cccccccccccccccccccc\}0&1\\\\ 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 $\displaystyle \sigma\_J$ 在 $\displaystyle V$ 的基 $\displaystyle E\_\{11\},\cdots,E\_\{22\}$ 下的矩阵为
\begin\{aligned\} D=\left(\begin\{array\}\{cccccccccccccccccccc\}0&0&1&0\\\\ -1&0&0&1\\\\ 0&0&0&0\\\\ 0&0&-1&0\end\{array\}\right)\sim \tilde\{C\}=C. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
易知 $\displaystyle D$ 的特征值全为 $\displaystyle 0$, 而 $\displaystyle D$ 幂零, $\displaystyle C$ 幂零, 选填’一定是‘.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](
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---
1513、 6、 (20 分) 设 $\displaystyle A$ 是数域 $\displaystyle \mathbb\{P\}$ 上的 $\displaystyle m\times n$ 矩阵, $\displaystyle \beta$ 是 $\displaystyle m$ 维非零列向量. 令
\begin\{aligned\} W=\left\\{\alpha\in\mathbb\{P\}^n; \exists\ t\in\mathbb\{P\},\mathrm\{ s.t.\} A\alpha=t\beta\right\\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
(1)、 (5 分) 证明: $\displaystyle W$ 关于 $\displaystyle \mathbb\{P\}^n$ 的运算构成 $\displaystyle \mathbb\{P\}^n$ 的子空间; (2)、 (15 分) 设线性方程组 $\displaystyle AX=\beta$ 的增广矩阵的秩为 $\displaystyle r$, 证明: $\displaystyle W$ 的维数等于 $\displaystyle n-r+1$. [张祖锦需要擅自在’增广矩阵‘前面加上’系数矩阵和‘才能做哦.) (中国矿业大学(徐州)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 $\displaystyle \forall\ k,k'\in\mathbb\{P\}, \alpha,\alpha'\in W$,
\begin\{aligned\} &\exists\ t,t'\in\mathbb\{P\},\mathrm\{ s.t.\} A\alpha=t\beta, A\alpha'=t'\beta\\\\ \Rightarrow&A(k\alpha+k'\alpha')=kA\alpha+k'A\alpha'=kt\beta+k't'\beta=(kt+k't')\beta\\\\ \Rightarrow&k\alpha+k'\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\}
故 $\displaystyle W$ 是 $\displaystyle \mathbb\{P\}^n$ 的子空间. (2)、 设 $\displaystyle \alpha\_1,\cdots,\alpha\_\{n-r\}$ 是 $\displaystyle AX=0$ 的一个基础解系, $\displaystyle \gamma\neq 0$ 是 $\displaystyle AX=\beta$ 的一个特解, 则由
\begin\{aligned\} &\sum\_i x\_i\alpha\_i+y\gamma=0\stackrel\{A\cdot\}\{\Rightarrow\}y\beta=0\stackrel\{\beta\neq 0\}\{\Rightarrow\}y=0 \Rightarrow \sum\_i x\_i\alpha\_i=0\Rightarrow x\_i=0,\ \forall\ i \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle \alpha\_1,\cdots,\alpha\_\{n-r\},\gamma$ 线性无关. 由
\begin\{aligned\} A\alpha\_i=0=0\beta,\quad A\gamma=\beta=1\cdot \beta \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle \alpha\_1,\cdots,\alpha\_\{n-r\},\gamma\in W$. 对 $\displaystyle \forall\ \alpha\in W$,
\begin\{aligned\} &\exists\ t\in \mathbb\{P\},\mathrm\{ s.t.\} A\alpha=t\beta\\\\ \Rightarrow&\left\\{\begin\{array\}\{llllllllllll\}\exists\ x\_i\in\mathbb\{P\},\mathrm\{ s.t.\} \alpha=\sum\_i x\_i\alpha\_i, &t=0,\\\\ A\frac\{\alpha\}\{t\}=\beta\Rightarrow \frac\{\alpha\}\{t\}-\beta=\sum\_i x\_i\alpha\_i \Rightarrow \alpha=\sum\_i tx\_i\alpha\_i+t\beta,&t\neq 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 $\displaystyle \alpha\_1,\cdots,\alpha\_\{n-r\},\gamma$ 是 $\displaystyle W$ 的一组基, $\displaystyle \dim W=n-r+1$.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](
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---
1514、 (3)、 若把同构的子空间称作一类, 则 $\displaystyle n$ 维线性空间的子空间共分成 $\displaystyle n$ 类. (中国人民大学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) / $\displaystyle \times$. $\displaystyle n$ 维线性空间 $\displaystyle V$ 的子空间的维数可能为 $\displaystyle 0,1,2,\cdots,n$, 共有 $\displaystyle n+1$ 类!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)/
---
1515、 (4)、 设 $\displaystyle V\_1,V\_2$ 是 $\displaystyle \mathbb\{P\}^n$ 的子空间, 且 $\displaystyle \dim V\_1+\dim V\_2=n$, 则 $\displaystyle \mathbb\{P\}^n=V\_1\oplus V\_2$. (中国人民大学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 $\displaystyle n=2$, $\displaystyle V\_1=V\_2=L(e\_1)$, 则 $\displaystyle \dim V\_1+\dim V\_2=2$, 但 $\displaystyle V\_1\cap V\_2=V\_1\neq \left\\{0\right\\}$, 而 $\displaystyle V\_1+V\_2$ 不是直和.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)/
---
1516、 (3)、 集合
\begin\{aligned\} V=\left\\{(x\_1,x\_2+\mathrm\{ i\} x\_3, -\mathrm\{ i\} x\_3, -x\_1)^\mathrm\{T\}, x\_1,x\_2,x\_3\in\mathbb\{R\}\right\\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
对于向量的加法和数乘构成实数域 $\displaystyle \mathbb\{R\}$ 上的线性空间, 则 $\displaystyle V$ 的一组基为 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (中国人民大学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\} \left(\begin\{array\}\{cccccccccccccccccccc\}x\_1\\\\x\_2+\mathrm\{ i\} x\_3\\\\ -\mathrm\{ i\} x\_3\\\\ -x\_1\end\{array\}\right)=x\_1\left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\0\\\\0\\\\-1\end\{array\}\right)+x\_2\left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\1\\\\0\\\\0\end\{array\}\right)+x\_3\left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\\mathrm\{ i\}\\\\-\mathrm\{ i\}\\\\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\}
及 $\displaystyle \left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\0\\\\0\\\\-1\end\{array\}\right), \left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\1\\\\0\\\\0\end\{array\}\right), \left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\\mathrm\{ i\}\\\\-\mathrm\{ i\}\\\\0\end\{array\}\right)$ 线性无关知 $\displaystyle V$ 的一组基为
\begin\{aligned\} \left(\begin\{array\}\{cccccccccccccccccccc\}1\\\\0\\\\0\\\\-1\end\{array\}\right), \left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\1\\\\0\\\\0\end\{array\}\right), \left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\\mathrm\{ i\}\\\\-\mathrm\{ i\}\\\\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\}
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)/
---
1517、 (4)、 已知 $\displaystyle (I)$ 和 $\displaystyle (I)'$ 是三维线性空间 $\displaystyle V$ 的两组基, $\displaystyle V$ 中的任意向量 $\displaystyle \gamma$ 在这两组基下的坐标 $\displaystyle (x\_1,x\_2,x\_3)$ 和 $\displaystyle (x\_1',x\_2',x\_3')$ 满足
\begin\{aligned\} x\_1'=x\_1, x\_2'=x\_2-x\_1, x\_3'=x\_3-x\_2. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
则 $\displaystyle (I)$ 到 $\displaystyle (I)'$ 的过渡矩阵是 $\displaystyle \underline\{\ \ \ \ \ \ \ \ \ \ \}$. (中国人民大学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 $\displaystyle (I): \varepsilon\_1,\varepsilon\_2,\varepsilon\_3$, $\displaystyle (II): \varepsilon\_1', \varepsilon\_2',\varepsilon\_3'$, 则
\begin\{aligned\} \gamma=&(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)\left(\begin\{array\}\{cccccccccccccccccccc\}x\_1\\\\x\_2\\\\x\_3\end\{array\}\right)\\\\ =&(\varepsilon\_1',\varepsilon\_2',\varepsilon\_3')\left(\begin\{array\}\{cccccccccccccccccccc\}x\_1'\\\\x\_2'\\\\x\_3'\end\{array\}\right) =(\varepsilon\_1',\varepsilon\_2',\varepsilon\_3')\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ -1&1&0\\\\ 0&-1&1\end\{array\}\right)\left(\begin\{array\}\{cccccccccccccccccccc\}x\_1\\\\x\_2\\\\x\_3\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\} (\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)=&(\varepsilon\_1',\varepsilon\_2',\varepsilon\_3')\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ -1&1&0\\\\ 0&-1&1\end\{array\}\right),\\\\ (\varepsilon\_1',\varepsilon\_2',\varepsilon\_3')=&(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ -1&1&0\\\\ 0&-1&1\end\{array\}\right)^\{-1\} =(\varepsilon\_1,\varepsilon\_2,\varepsilon\_3)\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ 1&1&0\\\\ 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\}
故应填 $\displaystyle \left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0\\\\ 1&1&0\\\\ 1&1&1\end\{array\}\right)$.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](
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---
1518、 4、 设有 $\displaystyle \mathbb\{R\}^4$ 的两个子空间
\begin\{aligned\} W\_1=&\left\\{(x\_1,x\_2,x\_3,x\_4)^\mathrm\{T\}; x\_1+2x\_2-x\_4=0\right\\}, \\\\ W\_2=&L(\alpha\_1,\alpha\_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 $\displaystyle \alpha\_1=(1,-1,0,1)^\mathrm\{T\}, \alpha\_2=(1,0,2,3)^\mathrm\{T\}$. 求 $\displaystyle W\_1+W\_2$ 和 $\displaystyle W\_1\cap W\_2$ 的基和维数. (中国人民大学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) / 取 $\displaystyle x\_2,x\_3,x\_4$ 为自由变量知
\begin\{aligned\} W\_1=L(\alpha\_3,\alpha\_4,\alpha\_5), \alpha\_3=\left(\begin\{array\}\{cccccccccccccccccccc\}-2\\\\1\\\\0\\\\0\end\{array\}\right), \alpha\_4=\left(\begin\{array\}\{cccccccccccccccccccc\}0\\\\0\\\\1\\\\0\end\{array\}\right), \alpha\_5=\left(\begin\{array\}\{cccccccccccccccccccc\}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\}
由
\begin\{aligned\} A&=(\alpha\_1,\cdots,\alpha\_5)=\left(\begin\{array\}\{cccccccccccccccccccc\}1&1&-2&0&1\\\\ -1&0&1&0&0\\\\ 0&2&0&1&0\\\\ 1&3&0&0&1\end\{array\}\right) \to\left(\begin\{array\}\{cccccccccccccccccccc\}1&0&0&0&-\frac\{1\}\{2\}\\\\ 0&1&0&0&\frac\{1\}\{2\}\\\\ 0&0&1&0&-\frac\{1\}\{2\}\\\\ 0&0&0&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\}
知 $\displaystyle W\_1+W\_2$ 有一组基 $\displaystyle \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4$, 且
\begin\{aligned\} \alpha\_5=-\frac\{1\}\{2\}\alpha\_1+\frac\{1\}\{2\}\alpha\_2-\frac\{1\}\{2\}\alpha\_3-\alpha\_4. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
从而 $\displaystyle \dim(W\_1+W\_2)=4$. 再者, 由
\begin\{aligned\} \alpha\in W\_1\cap W\_2&\Leftrightarrow \alpha=x\_1\alpha\_1+x\_2\alpha\_2=-x\_3\alpha\_3-x\_4\alpha\_4-x\_5\alpha\_5\\\\ &\Leftrightarrow \alpha=x\_1\alpha\_1+x\_2\alpha\_2, Ax=0\\\\ &\Leftrightarrow \alpha=x\_1\alpha\_1+x\_2\alpha\_2, x=k\left(\frac\{1\}\{2\},-\frac\{1\}\{2\},\frac\{1\}\{2\},1,1\right)^\mathrm\{T\}\\\\ &\Leftrightarrow \alpha=k\left(\frac\{1\}\{2\}\alpha\_1-\frac\{1\}\{2\}\alpha\_2\right) =-\frac\{k\}\{2\}(\alpha\_2-\alpha\_1)=-\frac\{k\}\{2\}(0,1,2,2)^\mathrm\{T\} \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
知 $\displaystyle W\_1\cap W\_2$ 有一组基 $\displaystyle (0,1,2,2)^\mathrm\{T\}$, $\displaystyle \dim (W\_1\cap W\_2)=1$.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](
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发表于 2023-3-5 13:24:56
