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Merge pull request #645 from JoinB-AI/patch-2
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Update chapter08.md
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jiangzhonglian authored Nov 30, 2023
2 parents cc69360 + cf91f4b commit ba390c9
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19 changes: 7 additions & 12 deletions docs/linalg/chapter08.md
Original file line number Diff line number Diff line change
Expand Up @@ -36,19 +36,16 @@ $$
讨论$b$满足什么条件才能让方程$Ax=b$有解(solvability condition on b):当且仅当$b$属于$A$的列空间时。另一种描述:如果$A$的各行线性组合得到$0$行,则$b$端分量做同样的线性组合,结果也为$0$时,方程才有解。

解法:令所有自由变量取$0$,则有$
\Big\lbrace
\begin{eqnarray*}
\begin{cases}
x_1 & + & 2x_3 & = & 1 \\
& & 2x_3 & = & 3 \\
\end{eqnarray*}
\end{cases}
$
,解得
$
\Big\lbrace
\begin{eqnarray*}
,解得$
\begin{cases}
x_1 & = & -2 \\
x_3 & = & \frac{3}{2} \\
\end{eqnarray*}
\end{cases}
$
,代入$Ax=b$求得特解
$
Expand All @@ -59,13 +56,11 @@ x_p=
$。

令$Ax=b$成立的所有解:

$$
\Big\lbrace
\begin{eqnarray}
\begin{cases}
A & x_p & = & b \\
A & x_n & = & 0 \\
\end{eqnarray}
\end{cases}
\quad
\underrightarrow{两式相加}
\quad
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