Why matrix of rank one can be written as a tensor product of two vectors? 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!

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 $\vec{b}^T = (b_1, \dots, b_N)$.
