Let be a matrix (over a field) in reduced row echelon form.
- Prove that the nonzero rows of are linearly independent. Prove that the pivotal columns are linearly independent and that each nonpivotal column is linearly dependent on the pivotal columns.
- Deduce that the row rank and column rank of are equal.
- We showed in this previous exercise that the nonzero rows of form a maximal linearly independent set amon the rows of . In particular, the row rank of is the number of pivots. Note that the pivotal columns of are elements of the standard basis on the column space of , and thus are linearly independent. Moreover, any nonpivotal column has only zeros in the entries corresponding to the nonpivotal rows of , and thus is linearly dependent on the pivotal columns.
- That is, the row rank and column rank of are both equal to the number of pivots of .