Matrix/Row rank and column rank/Fact/Proof

Proof

In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. The row rank of equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank , since the first rows are linearly independent, and, apart from this, this, there are only zero rows. It has also column rank , since the columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these columns. By exercise, the column rank is preserved by elementary row manipulations.