What is the importance of Row Space? Why is the row space of a  matrix important?I use it to define the rank of a matrix but are there other applications?
 A: Here is a possible interpretation (I'm not sure this at the appropriate level though):
Let $V$ be a finite dimensional vector space. To every linear map $\varphi: V \to V$ we can associate a dual map $\varphi^\ast: V^\ast \to V^\ast$, given by mapping a linear functional $f\in V^\ast$ to $\varphi^\ast f$, where 
$$(\varphi^\ast f )(v) := (f\circ \varphi)(v) \qquad \forall v\in V$$
One readily checks that $\varphi^\ast$ is in fact a linear map. Now choose a basis $e_1, \dots, e_n$ for $V$ and let $e^1, \dots, e^n$ be the corresponding dual basis of $V^\ast$. It is then natural to ask the following question:

What is the matrix representation of $\varphi^\ast$ w.r.t. the basis $e^1, \dots, e^n$ of $V^\ast$? And how does it relate to the matrix representation of $\varphi$ w.r.t. the basis $e_1, \dots, e_n$ of $V$?

If $\varphi$ has matrix representation $A = (a_{ij})_{i,j}$, i.e. if $\varphi(e_j) = \sum_{i=1}^n a_{ij} e_i$, then for $1\le j,k \le n$
$$(\varphi^\ast e^j)(e_k) = e^j (\varphi(e_k)) = e^j\left(\sum_{i=1}^n a_{ik} e_i\right) = \sum_{i=1}^n a_{ik} e^{j}(e_i) = a_{jk} = \sum_{i=1}^n a_{ji}e^i (e_k)$$
From which it follows that $\varphi^\ast e^j = \sum_{i=1}^n a_{ji} e^i$, which is to say $\varphi^\ast$ has matrix representation $(a_{ji})_{i,j} = A^T$ w.r.t. the basis $e^1, \dots, e^n$.

So if we interpret your $A$ as a linear transformation w.r.t. the canonical basis on $\mathbb R^n$, then the column space gives you the image of $A$, while the row space gives you the image of the dual of $A$.

