# Schedule

• Complete solution of Ax=b
• Rank r
• $r=m$ : Solution & Exists
• $r=n$ : Solution is Unique

# Complete solution of Ax=b

x_1+2x_2 +2x_3+2x_4 = b_1\2x_1+4x_2+6x_3+8x_4=b_2\3x_1+6x_2+8x_3+10x_4=b_3

Argumented Matrix = [A |b]=\begin{bmatrix}\fbox1&2&2&2&b_1\2&4&6&8&b_2\3&6&8&10&b_3\end{bmatrix}=\begin{bmatrix}\fbox1&2&2&2&b_1\0&0&\fbox2&4&b_2-2b_1\0&0&2&4&b_3-3b_1\end{bmatrix}=\begin{bmatrix}\fbox1&2&2&2&b_1\0&0&\fbox2&4&b_2-2b_1\0&0&0&0&b_3-b_2-b_1 \end{bmatrix}

Argumented Matrix = [A |b]=\begin{bmatrix}\fbox1&2&2&2&1\0&0&\fbox2&4&3\0&0&0&0&0\end{bmatrix}

Solvability is the condition on b.

Ax=b is solvable if when exactly when b is in the column space of A.

The same combination of the entries of b must give 0(not zero row, but number 0).

Question Mark Here:

## Find complete solution to Ax=b – 求Ax=b的所有解

Step one : A particular solution.

Set all free variable to zero and then solve Ax=b for pivot variables.

x_{\text{particular solution}} = \begin{bmatrix}-2\0\1.5\0\end{bmatrix}

Step two : add on X anything out of the null space.
Step three : 从而求得x=x_p+ x_n

The complete solution is the one particular solution plus any vector out of the null space.

Ax_p = bAx_n=0 两者相加,同样得到A(x_p+x_n)=b

x_{complete} =\begin{bmatrix}-2\0\1.5\0\end{bmatrix} + c_1\begin{bmatrix}-2\1\0\0\end{bmatrix} + c_2 \begin{bmatrix}2\0\-2 \1\end{bmatrix}
x_p是一个特定解,x_n是整个零空间,

Ax=b特解表示为particular solution(特定解), Ax=0基为special solution(特殊解).

## $m\times n$ matrix $A$ of rank $r$ – 秩为$r$的$m\times n$矩阵

1. Full column rank means r=n\lt m
这种情况下每列均有一个主元,从而没有自由变量. 这样的话零空间中将会只有零向量.
那么对于Ax=b来说,其全部解为特解x_p一个(如果有解的话), 将其称为unique solution(唯一解).
也就是说,对于r=n的情况下,其解的情况为0或者1个解(特定解).
举个例子:

2. Full row rank means r=m\lt n
这种情况下每行均有一个主元,自由变量数为n-r个.
因为在消元过程中没有产生零行,所以求解Ax=b对于b来说没有要求(Can solve Ax=b for every right-hand side),所以必然有解.
举个例子(上个例子的转置):

A=\begin{bmatrix}1&2&6&5\3&1&1&1\end{bmatrix}(r=2)\rightarrow R= \begin{bmatrix}1&2&6&5\0&-5&-17&-14\end{bmatrix}

3. $r=m=n$
零空间中只有零向量.
举个例子:
A=\begin{bmatrix}1&2\3&1\end{bmatrix}\rightarrow R= I
必然有解,唯一解.

r=m=n\rightarrow R=I\rightarrow 1 solution(特定解)
r=n\lt m\rightarrow R=I/0\rightarrow 0 or 1 solution(特定解)
r=m\lt n\rightarrow R=[I|F]\rightarrow 1 or infinitely many solutions(特定解或特定解和零向量空间的组合)
r\lt m,r\lt n\rightarrow 0 or infinitely many solutions(特定解和零向量空间的组合)

The rand tells everything about the number of solutions .