Proving that same solution set implies row equivalence The question I am trying to solve is for a much simpler case:

Suppose $R$ and $R'$ are $2\times3$ row-reduced echelon matrices and that the system $RX=0$ and $R'X=0$ have exactly the same solutions. Prove that $R = R'$.

I am approaching the problem as such:
All $2\times3$ matrices in RREF are of the following:
\begin{bmatrix} a & b & c \\ 0 & d & e \end{bmatrix}
Let $$R=\begin{bmatrix} a & b & c \\ 0 & d & e \end{bmatrix}, ~R'=\begin{bmatrix} a' & b' & c' \\ 0 & d' & e' \end{bmatrix}, ~X=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$
Rearranging equations, we eventually have that:
$$
(a-a')x_1+(b-b')x_2+(c-c')x_3=0\\
(d-d')x_2+(e-e')x_3=0
$$
Clearly, this holds for the trivial case when $x_1=x_2=x_3=0$, but I have no idea how to proceed from here.
Also, I was wondering how the general case that same solution set implying row equivalence could be proven rigorously, since it was only given to me as a theorem without proof. I can easily understand how row equivalence implies identical solution sets, but I am unsure how the reverse could be done.

$A$ is row equivalent to $B$ iff rows of $A$ can be obtained by applying elementary row operations to those of $B$.
Hence, two matrices are row equivalent iff row space of both matrices is same.
In given problem, Null space of both matrices is given same and we know, Row space is orthogonal complement of null space.
So, row spaces will be same in given problem. 
