Prove that two row equivalent matrices over a field have the same row rank.
Let be a field.
Suppose and are row equivalent via the matrix , which is a product of elementary matrices. That is, , so that . Let and denote the standard bases, and say , , and . In particular, we have . By this previous exercise, the row rank of is the column rank of is the dimension of , and the row rank of is the column rank of is the dimension of . Since and is an isomorphism, we have , and likewise since , . Thus we have , so that and have the same row rank.