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: 
Let a matrix $A$ have columns $\vec{a}\_i$, $i = 1, \dots, N$. A rank one matrix is characterised by the fact that all columns $\vec{a}\_i$ are linearly dependent, that is
$$
\vec{a}\_i = b\_i \vec{a}
$$
for some vector $\vec{a}$ and scalars $b\_i$. Thus, the matrix $A$ takes the form
$$
A = (b\_1 \vec{a}, \dots, b\_N \vec{a}) = \vec{a} \vec{b}^T = \vec{a} \otimes \vec{b},
$$
where $\vec{b}^T = (b\_1, \dots, b\_N)$.
For example $$\begin{pmatrix}1&1&1\\2&2&2\\3&3&3\end{pmatrix}$$
 has rank one and is $(1,1,1)\otimes (1,2,3)$