I have shown that two of the three elementary operations will not change the image of the row space of the matrix: given a row vector $\vec{v}$, $k\vec{v}$ will span the same (scalar multiplication), and switching the order of the rows won't change anything because the span of the row vectors will be the same (vectors will be in a different order). However, how would one show that adding/subtracting vectors does not change the row space?
Linear Algebra – Why Elementary Matrix Operations Do Not Affect Row Space
linear algebramatrix-rank
Best Answer
Suppose the rows of the matrix are $A=(v_1 ,v_2 ,...,v_n)$, then switching two rows, is just switching the order of the vectors, and as you said, it doesn't change the span of this set. The same is true for multiplying by a scalar ($\neq 0$) one of the rows.
Suppose you want to change $v_2$ to the vector $v_2 + \alpha v_1$ which is the last elementary operation. The span of $B=(v_1 ,v_2 + \alpha v_1 ,...,v_n)$ will contain $(v_2 + \alpha v_1) - \alpha v_1 = v_2$ so $A \subseteq \operatorname{span}(B) \Rightarrow \operatorname{span}(A) \subseteq \operatorname{span}(B)$. In the same way $v_2 + \alpha v_1 \in \operatorname{span}(A)$ so $B\subseteq \operatorname{span}(A) \Rightarrow \operatorname{span}(B) \subseteq \operatorname{span} (A)$.
now you have that $\operatorname{span}(B)=\operatorname{span}(A)$ so adding one row (times a scalar) to another doesn't change the row space.