1. Let A be a m*n matrix with no row swaps during gaussian elimination
2. Each row transformation can be represented as a matrix (...E43....E32....E41E31*E21) as E since these row transformations are reversible which means the final matrix E is invertible and the columns of E are independent.
3. EA=U => A=E^-1U
4. Here U is NOT an upper triangular matrix but a matrix with all elements below the diagonal as zero but the diagonal elements themselves can be zero.
5. the diagonal elements of U which are zero can be written as a linear combination of columns of U making them dependent columns of U
6. Write the dependent columns of U as the linear combinations of the independent columns of U
7. Now when E^-1 is multiplied to U [E^-1 C1,E^-1C2,…..,E^-1Cn] the columns with non zero pivots remain unique columns in the final matrix and the dependent columns become dependent on the new independent columns of the final matrix and hence Column independence is retained in this transformation