Let $\mathbb{I}_{3}$ be the $3\times 3$ identity matrix, i. e.
$\begin{pmatrix} 1 &0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
and $e_1:=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $e_2:=\begin{pmatrix} 0 \\ 1 \end{pmatrix}$.
My question is, if the matrix $(\mathbb{I}_{3}\otimes e_1,\mathbb{I}_{3}\otimes e_2) $ is the same as $\mathbb{I}_{3}\otimes \mathbb{I}_{2}$? I have illustrated it, but do not know how to write this down mathematically.
$A\otimes B=\begin{pmatrix} a_{11} B & a_{12} B \\ a_{21} B & a_{22} B \end{pmatrix}$
It is
$(\mathbb{I}_3 \otimes e_1,\mathbb{I}_3 \otimes e_2)= \begin{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} & 0 & 0 \\ 0 & \begin{pmatrix} 1 \\ 0 \end{pmatrix} & 0 \\ 0 & 0 & \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{pmatrix}, \begin{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} & 0 & 0 \\ 0 & \begin{pmatrix} 0 \\ 1 \end{pmatrix} & 0 \\ 0 & 0 & \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{pmatrix}$
But i don’t see in it $\mathbb{I}_{3}\otimes \mathbb{I}_{2}$.