I am trying to work out some problems and then I realise something funny. Say I have a matrix : $ A=\begin{bmatrix} -2 & 1 & 3\\ -1.5 & 1 & 2\\ -1.5 & 1 & 2 \end{bmatrix}$
The $\operatorname{rank}(A)=2$. I want to find the null space of the matrix $A$: $ \begin{bmatrix} -2 & 1 & 3\\ -1.5 & 1 & 2\\ -1.5 & 1 & 2 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}=0$
And I got it through the usual steps: $ N(A)=t\begin{bmatrix} 2\\ 1\\ 1 \end{bmatrix}, t \in \mathbb{R}$
But what surprises me is that the null space could be a combination of the columns of $A$! $ \begin{bmatrix} 2\\ 1\\ 1 \end{bmatrix}=-2\begin{bmatrix} -2 \\ -1.5 \\ -1.5 \end{bmatrix}-2 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$
How is this possible? I thought if it in the null space, then it wouldn't be in the column space of $A$.