Let $$A=\left[\begin{array}{ccc} 9 &-9 &-3\cr -3 &3 &1 \end{array}\right].$$
Find bases for the kernel and image of $A$ (or, equivalently, for the linear transformation $T(x)=Ax$).

A basis for the kernel of $A$ 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.$ $\Bigg\rbrace.$

A basis for the image of $A$ is $\Bigg\lbrace$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ $\Bigg\rbrace.$