Answer
We have two free variables $x_{3},x_{5}$. $\mathbf{dimension\,of\,Nul\,A\,is\,2}$
we have three pivotal columns, $\mathbf{dimension\,of\,Col\,A\,is \,3}$
Work Step by Step
We are required to determine the dimensions of Nul A and Col A for the given matrix.
$\mathbf{A}=\begin{bmatrix}1& -6& 9& 0& -2&\\0& 1& 2& -4& 5\\0 &0 &0 &5&1\\0&0&0&0&0\end{bmatrix}$
Row reducing the augmented matrix to echelon form:
$\mathbf{A}=\begin{bmatrix}1& -6& 9& 0& -2&\\0& 1& 2& -4& 5\\0 &0 &0 &5 &1\\0&0&0&0&0\end{bmatrix}\sim\begin{bmatrix}1& 0& 21& 0& \frac{164}{5}&0\\0& 1& 2& 0& \frac{29}{5}&0\\0 &0 &0 &1&\frac{1}{5}&0\\0&0&0&0&0&0\end{bmatrix}$
The dimension of Nul A is the number of free variables in the equation [Ax=0]
We have two free variables $x_{3},x_{5}$. $\mathbf{dimension\,of\,Nul\,A\,is\,2}$
The dimension of Col A is the number of pivot columns in A.
we have three pivotal columns, $\mathbf{dimension\,of\,Col\,A\,is \,3}$
