# （6）列空间和零空间

$$A$$的列空间由所有列的线性组合构成
columan space of $$A$$ is subspace of $$R^{4}$$ = all linear combination of columns

$$Ax: \begin{bmatrix}1&1&2\\2&1&3\\3&1&4 \\4&1&5\end{bmatrix} \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}=\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\b_{4}\end{bmatrix}$$

Nullspace of $$A$$ = all solutions $$x=\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}$$ to $$Ax=0$$

$$Ax= \begin{bmatrix}1&1&2\\2&1&3\\3&1&4 \\4&1&5\end{bmatrix} \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}$$

$$N(A)$$ contains $$\begin{bmatrix}0\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\-1\end{bmatrix}$$