Why are elementary row operation linear transformation? I just started to learn linear algebra and found out that elementary operations could be written in form of matrix so it means that elementary row operations are linear transformation but couldn't justify it. I understand how swap and scale elementary row operations are linear transformation but couldn't understand why sRi + Rj --> Rj is a linear transformation.
 A: Let's concentrate on a $3\times 3$ matrix for now. The general theory will become clear. The core idea is that row operations correspond to multiplying by a particular matrix.
Given a matrix
$$
M \equiv \begin{bmatrix}
v_1^1 &  v_1^2 &  v_1^3 \\
v_2^1 &  v_2^2 &  v_2^3 \\
v_3^1 &  v_3^2 &  v_3^3 \\
\end{bmatrix}
$$
We can write our matrix 
$M \equiv \begin{bmatrix} r_1 \\ r_2 \\ r_3 \end{bmatrix}$ where each $r_i$ are the rows, defined as $r_i \equiv  \begin{bmatrix} v_i^1  & v_i^2 &  v_i^3 \end{bmatrix}$.
Now, we can look at the row transformation $R_1 \rightarrow \alpha R_1 + \beta R_2+ \gamma R_3$ on matrix $M$ to yield matrix $M'$ as:
$$
\begin{align*}
M' &\equiv 
\begin{bmatrix}
\alpha & \beta & \gamma \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
M \\
&=
\begin{bmatrix}
\alpha & \beta & \gamma \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix} r_1 \\ r_2 \\ r_3 \end{bmatrix} \\
&=
\begin{bmatrix} \alpha r_1 + \beta r_2 + \gamma r_3 \\ r_2 \\ r_3 \end{bmatrix}
\end{align*}
$$
So, the matrix $M'$ (obtained after a row transformation) is a linear transform applied onto the original matrix $M$.
For a general transformation, we can create the transformation matrix appropriately, generalising from this example.
This explains why row transformations cannot change the span of the rows: all these transformations can do is to take combinations of existing rows, which does not allow one to access vectors outside the subspace spanned by $\{ r_1, r_2, r_3 \}$.
(note: you might want to check that it's indeed legal to collapse a matrix into the rows $r_i$, and that the composition rules do work out. They do, but it's a good exercise to check that writing the matrix as $r_i$ and performing transformations is the same as writing the entire $v_i^j$.)
