If B is invertible matrix , then the Row space of A and BA are same I have seen a proof that  B is row equivalent to A iff exist invertible matrix C such that :
B = CA , because C is elementary matrix  , but i  cant find the next step . can B "used" as the elementary matrix that lead from B to BA without changing the row space of A  , so A and BA have same rowspace? 

Well, since $B=CA$, you see that rows of $B$ can be written as linear combinations of rows of $A$. (To be more explicit, write 
$$B=\begin{bmatrix}\mathbf{b_1\\b_2\\ \vdots \\ b_m}\end{bmatrix},\quad A=\begin{bmatrix}\mathbf{a_1\\a_2\\ \vdots, \\ a_m}\end{bmatrix},\quad C=\begin{bmatrix}c_{11} &\cdots & c_{1m}\\
\vdots & \ddots&\vdots\\ c_{m1} & \cdots & c_{mm}
\end{bmatrix}$$
and perform partitioned matrix multiplication. Here letters in boldface denote row vectors comprising matrices.)
Likewise we note $A=C^{-1}B$.
