Answer
We have three free variables $Column\,{2},Column\,{5},Column\,{6}$. $\mathbf{dimension\,of\,Nul\,A\,is\,3}$
we have three pivotal columns $Column\,{1},Column\,{3},Column\,{4}$, $\mathbf{dimension\,of\,Col\,A\,is \,3}$
Work Step by Step
We are required to determine the dimensions of Nul A and Col A for the given matrix.
The given matrix is already row reduced to echelon form:
$\mathbf{A}=\begin{bmatrix}1&3&{ - 4}&2&{ - 1}&6\\0&0&1&{ - 3}&7&0\\0&0&0&1&4&{ - 3}\\0&0&0&0&0&0\end{bmatrix}$
The dimension of Nul A is the number of free variables in the equation [Ax=0]
We have three free variables $Column\,{2},Column\,{5},Column\,{6}$. $\mathbf{dimension\,of\,Nul\,A\,is\,3}$
The dimension of Col A is the number of pivot columns in A.
we have three pivotal columns $Column\,{1},Column\,{3},Column\,{4}$, $\mathbf{dimension\,of\,Col\,A\,is \,3}$
