张祖锦常用结论05维数公式

zhangzujin

手机查看请在浏览器中打开, 到了支付页面请截图, 并用支付宝或微信扫描之, 稍等后获得金钱, 即可购买. 偶偶因为网络问题充值不成功, 请与微信 pdezhang 联系, 发送论坛昵称与付款时间即可处理, 稍安勿躁. 购买后刷新网页才能正常显示数学公式.

# 张祖锦常用结论维数公式 $\displaystyle \dim \ker\mathscr\{A \}+\dim \mathrm\{im\} \mathscr\{A\}=n$ 的证明过程非常有用 设 $\displaystyle U,V$ 是数域 $\displaystyle \mathbb\{F\}$ 上的线性空间, $\displaystyle \dim U=n$, $\displaystyle \mathscr\{A\}: U\to V$ 是线性映射, 则 \begin\{aligned\} \dim \ker \mathscr\{A\}+\dim \mathrm\{im\} \mathscr\{A\}=n. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} [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 $\displaystyle U$ 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 \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)/