The row rank and column rank of a reduced row echelon matrix are equal

Let A be a matrix (over a field) in reduced row echelon form.

  1. Prove that the nonzero rows of A are linearly independent. Prove that the pivotal columns are linearly independent and that each nonpivotal column is linearly dependent on the pivotal columns.
  2. Deduce that the row rank and column rank of A are equal.

  1. We showed in this previous exercise that the nonzero rows of A form a maximal linearly independent set amon the rows of A. In particular, the row rank of A is the number of pivots. Note that the pivotal columns of A are elements of the standard basis on the column space of A, and thus are linearly independent. Moreover, any nonpivotal column has only zeros in the entries corresponding to the nonpivotal rows of A, and thus is linearly dependent on the pivotal columns.
  2. That is, the row rank and column rank of A are both equal to the number of pivots of A.
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