Answer
We have one free variable $Column\,{3}$. $\mathbf{dimension\,of\,Nul\,A\,is\,1}$
We have two pivotal columns $Column\,{1},Column\,{2}\,\mathbf{dimension\,of\,Col\,A\,is \,2}$
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 } & { 4 } & { - 1 } \\ { 0 } & { 7 } & { 0 } \\ { 0 } & { 0 }&0\end{bmatrix} $
The given matrix is already in echelon form.
The dimension of Nul A is the number of free variables in the equation$ [AX=0]$
We have one free variables $Column\,{3}$. $\mathbf{dimension\,of\,Nul\,A\,is\,1}$
The dimension of Col A is the number of pivot columns in A.
we have two pivotal columns $Column\,{1},Column\,{2}\,\mathbf{dimension\,of\,Col\,A\,is \,2}$