Vector space/Chain/Flag/2/Exercise

Determine whether the following chain of linear subspaces in ${}\mathbb {R} ^{3}$ is a flag.

$V_{0}=0,\,V_{1}=\mathbb {R} {\begin{pmatrix}8\\-3\\1\end{pmatrix}},$

$V_{2}=\operatorname {kern} \varphi ,{\text{ where }}\varphi :\mathbb {R} ^{3}\rightarrow \mathbb {R} {\text{ is given by the matrix}}\left(1,\,2,\,-2\right),\,V_{3}=\mathbb {R} ^{3}.$

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