Matrix/Row rank and column rank/Manipulations/Fact/Proof

Proof

Let ${}r$ denote the number of the relevant rows in the matrix ${}M'$ in echelon form, gained by elementary row manipulations. We have to show that this number is the column rank, and the row rank of ${}M'$ and of ${}M$. In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. So the row rank of ${}M$ equals the row rank of ${}M'$. This matrix has row rank ${}r$, since the first ${}r$ rows are linearly independent, and beside this, there are only zero rows. But ${}M'$ has also 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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