Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier.

$A=\left[\begin{array}{ccc} 1 &2 &12\cr 3 &3 &21\cr 2 &6 &34 \end{array}\right] \sim \left[\begin{array}{ccc} 1 &0 &2\cr 0 &1 &5\cr 0 &0 &0 \end{array}\right]$.

A basis for the column space 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 row space of $A$ is
$\Bigg\lbrace$$\left[\Rule{0pt}{1.2em}{0pt}\right.$$\left]\Rule{0pt}{1.2em}{0pt}\right.$, $\left[\Rule{0pt}{1.2em}{0pt}\right.$$\left]\Rule{0pt}{1.2em}{0pt}\right.$$\Bigg\rbrace$.

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