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 $M$ equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank $r$, since the first $r$ rows are linearly independent, and, apart from this, there are only zero rows. It also has column rank $r$, since the $r$ columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these $r$ columns. By exercise, the column rank is preserved by elementary row manipulations.

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