Let $$A = \left[\begin{array}{cccc} 0 &-1 &-2 &-1\cr -4 &0 &4 &4 \end{array}\right].$$ Find orthonormal bases of the kernel, row space, and image (column space) of $A$.

(a) Basis of the kernel:
$\Bigg\lbrace$ $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$, $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$ $\Bigg\rbrace.$

(b) Basis of the row space:
$\big\lbrace$ $\left[\Rule{0pt}{1.2em}{0pt}\right.$$\left]\Rule{0pt}{1.2em}{0pt}\right.$, $\left[\Rule{0pt}{1.2em}{0pt}\right.$$\left]\Rule{0pt}{1.2em}{0pt}\right.$ $\big\rbrace.$

(c) Basis of the image (column space):
$\Bigg\lbrace$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$, $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ $\Bigg\rbrace.$