Schedule

• Computing the null-space (Ax=0)
• Pivot variable with Free variable
• Special Solutions — rref(A)=R
这章主要讨论的是长方矩阵(rectangular matrix)

Computing the Nullspace – 计算零空间

A=\begin{bmatrix}1&2&2&2\2&4&6&8\3&6&8&10\end{bmatrix}

A=\begin{bmatrix}\fbox1&2&2&2\2&4&6&8\3&6&8&10\end{bmatrix}=\begin{bmatrix}1&2&2&2\0&0&2&4\0&0&2&4\end{bmatrix}

A=\begin{bmatrix}\fbox1&2&2&2\2&4&6&8\3&6&8&10\end{bmatrix}=\begin{bmatrix}\fbox1&2&2&2\0&0&\fbox2&4\0&0&2&4\end{bmatrix}=\begin{bmatrix}\fbox1&2&2&2\0&0&\fbox2&4\0&0&0&0\end{bmatrix}=U

Rank of A = # of pivots

\begin{bmatrix}-2\1\0\0\end{bmatrix} 和 \begin{bmatrix}2\0\-2\1\end{bmatrix}

The null space contains exactly all the combinations of the special solutions.

There is one special solution for every free variable.

R = reduced row echelon form
U=\begin{bmatrix}\fbox1&2&2&2\0&0&\fbox2&4\0&0&0&0\end{bmatrix}=\begin{bmatrix}\fbox1&2&0&-2\0&0&\fbox2&4\0&0&0&0\end{bmatrix}=\begin{bmatrix}\fbox1&2&2&2\0&0&\fbox1&2\0&0&0&0\end{bmatrix}=R=rref(A)

notice \begin{bmatrix}1&0\0&1\end{bmatrix} = I in pivot rows and pivot column.

rref(A)中的全零行表示该行原为其他行的线性组合,可以被消元过程中去除.

x_1+2x_2-2x_4=0\\x_3+2x_4=0

$rref$ – 简化行阶梯形式

R=\begin{bmatrix}I&F\0&0\end{bmatrix}

N = \text{null-space matrix(columns of special solutions)}=\begin{bmatrix}-F\I\end{bmatrix}