Can anyone please tell me why the number of linearly independent columns of a matrix doesn't change even if we apply row operations on the matrix?
The column space does change by row operations but I don't understand why the number of independent columns remains the same.
Row operations change columns and change the column space, but they preserve linear dependence relations among the columns!
For example, say $C_1$ and $C_2$ are the first two columns, which become $C_j'$ after a row operation. You can verify that if $aC_1+bC_2=0$ then $aC_a'+bC_2'=0$.
Hence a set of columns is a basis for the column space if and only if it is a basis after the row operation.