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 $M$, $rank(M) = min\{ r \vert \forall i \in \{1,2..,m\}, j \in \{1,2,..,n\} \exists v_i , w_j \in \mathbb{R}^r , M_{ij} = v_i^T w_j \}$
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.