Let and be row equivalent matrices over a field, say by an invertible matrix such that .
- Prove that, for all matrices (such that the dimensions match), if and only if .
- Prove that any linear dependence satisfied by the columns of is also satisfied by the columns of .
- Conclude that and have the same column rank.
- If , then , so . Conversely, if , then , so .
- Write and as column matrices: and . Note that, since , we have . In particular, for all . Now suppose is a linear dependence among the columns of . Now apply (as a linear transformation) to this equation; we have . Thus the columns of satisfy the same linear dependence.
- Suppose is a maximal linearly independent set of columns of . By part (2), is a linearly independent set of columns of . If this set is not maximal, then by part (2) again (using ) is not maximal, a contradiction. So is a maximal linearly independent set. Since is injective (as a linear transformation), these sets have the same cardinality. In particular, and have the same column rank.