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8.08 可解性和解的结构 #1

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morrQAQ opened this issue Jul 4, 2023 · 0 comments
Open

8.08 可解性和解的结构 #1

morrQAQ opened this issue Jul 4, 2023 · 0 comments

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@morrQAQ
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morrQAQ commented Jul 4, 2023

可解性的解释

Ax=b solvable when b is in column space of a

方程组(行空间)角度可以用平行来理解,空间变换(列空间)角度可以用向量的线性组合来理解.  
$$ \begin{bmatrix} a&b\\\ c&d\\\ \end{bmatrix} \times \begin{bmatrix} x\\\ y\\\ \end{bmatrix} = x\begin{bmatrix} a\\\ c\\\ \end{bmatrix} + y\begin{bmatrix} b\\\ d\\\ \end{bmatrix} = \begin{bmatrix} ax+by\\\ cx+dy\\\ \end{bmatrix} $$
这是一个超级屌的等式.
可以看出,列空间是所有(x * 列向量+y*列向量+...)的向量和,而它们等于b.  
用二维举例就是两个向量的线性组合等于b.  

if the combination of rows gives the zero row,b either

0+0=0
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