I'm asked to demonstrate that all the matrices $n*m$ with elements in K and $rank \le 1 $ are in the form $$\begin{pmatrix} a_1 \\ a_2\\...\\ a_m \end{pmatrix} \begin{pmatrix} b_1 & b_2 & ... & b_n \end{pmatrix} $$ with $a_1, a_2,...a_m, b_1 , b_2, ..., b_n \in K $
My attempt is:
A matrix $n*m$ with $rank \le 1 $ can be reduced to a row vector or a column vector through a series of elementary operations with its rows and columns.
Is it right?
You are right, but let me present you another interpretation of $\mathrm{rank}(M)\leq 1$:
It also means that the columns of the matrix span a subspace of at most dimension $1$, so all the columns are multiples of a single vector $\vec a=(a_1\ a_2\; \cdots\; a_m)^\top$. The factor by which each columns is a multiple of $\vec a$ are collected in your row-vector $\vec b=(b_1\;b_2\;\cdots\;b_n)$. Thats basically what your construction expresses: constructing a matrix where column $i$ is a $\vec a$ times $b_i$.