Find bases of the kernel and image of the orthogonal projection onto the plane $3 x + 3 y + z = 0$ in ${\mathbb R}^3$.

A basis for the kernel is $\Bigg\lbrace$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\Bigg\rbrace.$

A basis for the image is $\Bigg\lbrace$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\Bigg\rbrace.$