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!

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.
