Row rank ($\dim \mathrm{row}(A)$)

The row rank of a matrix is the dimension of the vector space spanned by its rows. Elementary row operations preserve this space; multiplication on the right by an invertible matrix preserves all linear relations among the rows. In row echelon form the nonzero rows are linearly independent, so the row rank is the number of pivots. The pivot columns are also linearly independent and span the column space, proving equality with the column rank.