张祖锦常用结论05维数公式
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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)/(https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/(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)/
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