About a definition of "rank" of a matrix. I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix.  
Another alternative definition is that given a $m \times n$ matrix $M$, $rank(M) = min\{ r \vert \forall i \in \{1,2..,m\}, j \in \{1,2,..,n\} \exists v_i , w_j \in \mathbb{R}^r ,  M_{ij} = v_i^T w_j  \}$


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*Can someone help show that the above rank means the usual thing? 

*Given a $M$ how does one find these needed $m+n$, $v$ and $w$ vectors? 

$rank(M) = \min\{r|\exists V \in L(m,r), W \in L(n,r). M = V^TW \}$ where $L(m,n)$ is the space of $n\times m$ matrices is my rewriting of the (intended) meaning of your definition.  You should verify that it is equivalent.  I think this definition (particularly compared to your definition of rank as the dimension of the image) makes it much easier to answer your first question.  In particular, the matrix $W$ should be pretty clear.  As to your second question, how do you find the image of a matrix?  Also, if $W$ is clear to you, you can consider that $M^T = W^TV$, so the $V$ matrix can be found in the same way as the $W$ matrix.
