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: 
The matrix $A$ is matrix of some linear map $\mathcal{A} \in \{\mathbb R^n \to \mathbb R^N\}$. Having rank $1$ means that in some bases $\{a\_1,a\_2,...,a\_n\}$ and $\{b\_1,b\_2,...,b\_N\}$ of respective spaces its matrix is $$\pmatrix{ 1 & 0 & ... & 0 \\ 0 & 0 &... & 0 \\ ... & ... & ... & ... \\ 0 & 0 & ... & 0 }$$
and thus $\mathcal A = a\_1 \otimes b\_1$. This equality holds for any numeric representation of $\mathcal A, a\_1, b\_1$ including initial basis you wrote matrix $A$ in.
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.