Find bases for the column space, the row space, and the null space of the matrix $\displaystyle{A = {\left[\begin{array}{cccc} 1 &3 &-1 &1\cr 3 &12 &0 &5\cr 2 &12 &4 &6 \end{array}\right]}}$ You should verify that the Rank-Nullity Theorem holds. Basis for the column space of $\displaystyle{A = \Bigg\lbrace}$$\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$$\displaystyle{\Bigg\rbrace}$

Basis for the row space of $\displaystyle{A = \Bigg\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.$ $\displaystyle{ \Bigg\rbrace}$

Basis for the null space of $\displaystyle{A = \Bigg\{ }$$\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$, $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$$\displaystyle{\Bigg\}}$