Proving that two row equivalent matrices are having the same row space

We begin by considering that A and B are two row equivalent matrices, which means that there exists an invertible matrix P such that: B=PA .

A1= all the rows in A.

A2 = all the rows in B.

Let’s also assume we know that every row in B is a linear combination of the rows of A if the two matrices are row equivalent (This part can be proven with some algebra).

Therefore every vector in B1 is a member of Span(A1).
Span(B1) is the smallest vector space that contains B1. Therefore Span(B1) is contained or equal to Span(A1).

We can also say that A=(P^-1)B, so we can do the same process and say that Span(A1) is contained or equal to Span(B1). So Span(A1)=Span(B1).
which means that the row space of A is equal to the row space of B.

Is this correct? Any suggestions to improve? Thanks!

(Sorry for not using MathJax, its the first proof I wrote in this site)

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