# Showing the linear independence between two row equivalent matrices

I can prove that if A and B are row equivalent matrices, then the column vectors of A are linearly independent iff the column vectors of B are linearly independent.

However, does this result also hold for row vectors? That is, is it true that if A and B are row equivalent matrices, then the row vectors of A are linearly independent iff the row vectors of B are linearly independent? How exactly do you prove this?

I know how to prove that elementary row operations do not change the row space of a matrix, but I'm not sure if that's any use here.

Solution: Let $A$ be an $n$ by $m$ matrix. If we assume that the $n$ row vectors of $A$ are linearly independent, then they form a basis for the row space of $A$ since they span the row space by definition. So we know that the dimension of the rowspace of $A$ is $n$. Now $B$ also has $n$ row vectors, since elementary row operations do not change the rowspace of a matrix, then the row vectors of $B$ also span the same rowspace of $A$. Thus, the row vectors of $B$ also forms a basis for the common rowspace. Hence, the row vectors of $B$ are linearly independent.

To prove the converse, note that we can go from matrix $B$ to $A$ by using inverse elementary row operations, hence the same argument can be used.

Hint: If the rows of $A$ are linearly independent, they form a basis of the row space; the rows of $B$ span the same subspace.