Question: I find in Wikipedia that a matrix $A\in\mathbb{M}^{N\times n}$ has rank one if it can be written as the following tensor product

$$A=a\otimes b,$$ with $b\in \mathbb{R}^N$ and $a\in\mathbb{R}^n$.

I know that a matrix has rank one if one is the maximum number of linearly independent colums of the matrix.

Why a rank one matrix can be written in that way?

Thank you!

Answer: 
For example $$\begin{pmatrix}1&1&1\\2&2&2\\3&3&3\end{pmatrix}$$
 has rank one and is $(1,1,1)\otimes (1,2,3)$
Look at the first non-zero column of $A$. That is $b$. Then every other column of $A$ is a multiple $b$, since the rank of $A$ is $1$. $a$ consists of these proportionality constants, in order.