Why is it that when we determine the pivot columns of an m x n matrix $A$, the pivot columns form a basis for the $Range(A)$? I understand that the pivot columns are linearly independent (the reason why we chose them as the pivot columns), but how do we know that they also span the range of $A$?
Also, why is that we're choosing the columns of $A$ to be the vectors that form a basis for $Range(A)$ rather than the columns of $U = rref(A)$?
Best Answer
The pivot columns form a basis for the $Range(A)$ since they are linearly independent.
We're choosing the columns of $A$ to be the vectors that form a basis for $Range(A)$ rather than the columns of $U = rref(A)$ because $U$ is generally obtained by row operations which modify the column space.