Answer
We have two free variables $Column\,{2},Column\,{4}$. $\mathbf{dimension\,of\,Nul\,A\,is\,2}$
we have two pivotal columns $Column\,{1},Column\,{3}$., $\mathbf{dimension\,of\,Col\,A\,is \,2}$
Work Step by Step
We are required to determine the dimensions of Nul A and Col A for the given matrix.
$\mathbf{A}=\begin{bmatrix}{ 1 } & { 0 } & { 9 } & { 5 } \\ { 0 } & { 0 } & { 1 } & { - 4 }\end{bmatrix}$
Row reducing the matrix m to echelon form:
$\mathbf{A}=\begin{bmatrix}{ 1 } & { 0 } & { 9 } & { 5 } \\ { 0 } & { 0 } & { 1 } & { - 4 }\end{bmatrix}\sim\begin{bmatrix}1&0&0&{41}\\
0&0&1&{ - 4}\end{bmatrix}$
The dimension of Nul A is the number of free variables in the equation$ [AX=0]$
We have two free variables $Column\,{2},Column\,{4}$. $\mathbf{dimension\,of\,Nul\,A\,is\,2}$
The dimension of Col A is the number of pivot columns in A.
we have two pivotal columns $Column\,{1},Column\,{3}$., $\mathbf{dimension\,of\,Col\,A\,is \,2}$
