Answer
We have no free variables. $\mathbf{dimension\,of\,Nul\,A\,is\,0}$
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.
$A = \begin{bmatrix} { 3 } & { 4 } \\ { - 6 } & { 10 } \end{bmatrix}$
Row reducing the matrix to echelon form:
$A = \begin{bmatrix} { 3 } & { 4 } \\ { - 6 } & { 10 } \end{bmatrix}\sim\begin{bmatrix} { 1 } & { 0 } \\ { 0 } & { 1 } \end{bmatrix}$
The dimension of Nul A is the number of free variables in the equation$ [AX=0]$
We have no free variables. $\mathbf{dimension\,of\,Nul\,A\,is\,0}$
The dimension of Col A is the number of pivot columns in A.
we have two pivotal columns $Column\,{1},Column\,{2}$., $\mathbf{dimension\,of\,Col\,A\,is \,2}$
