null space of matrix

Jun 2019
8
0
Massachusetts
For the following image:

Screen Shot 2019-11-05 at 9.15.58 PM.png

It is not obvious to me that null(M) is the plane with equation x + 2y + 2z = 0. How do we know that it's a plane? Since there are 3 variables, aren't we dealing with a hyper-plane ?
 

romsek

MHF Helper
Nov 2013
6,666
3,004
California
it's directly from the definition.

The null space of A is all vectors $v = \begin{pmatrix}x\\y\\z\end{pmatrix} \ni Av=0$

$A = \begin{pmatrix}1 & 2 &2 \end{pmatrix}$

$\begin{pmatrix}1 &2 &2\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = 0 \Rightarrow x + 2y + 2z = 0$