We consider the
linear mapping
-
given by the matrix
-
![{\displaystyle {}M={\begin{pmatrix}0&1&1\\0&2&2\\1&3&4\\2&4&6\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40e2229bd2fa1b7e29a9767bd3baca9c64da1b8)
To determine the
kernel,
we have to solve the
homogeneous linear system
-
![{\displaystyle {}{\begin{pmatrix}y+z\\2y+2z\\x+3y+4z\\2x+4y+6z\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c22ed04aba59c1fb8d848a3787e4e5ab8673f49a)
The solution space is
-
![{\displaystyle {}L={\left\{s{\begin{pmatrix}1\\1\\-1\end{pmatrix}}\mid s\in \mathbb {R} \right\}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e841b6ff757287d738217986df968c4787223f2d)
and this is the kernel of
. The kernel has dimension one, therefore the dimension of the image is
, due to
the dimension formula.