Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis $\{e_i\otimes f_j\}$ of $\mathbb{R}^2\otimes\mathbb{R}^3$ where $x = (1, 1)$ and $y = (1, -2, 1)$.
Since I have that $(1, 1) = e_1 + e_2$ and $(1, -2, 2) = f_1 - 2f_2 + f_3$, then $x \otimes y = (1, 1) \otimes (1, -2, 1) = (e_1 + e_2)\otimes (f_1 - 2f_2 + f_3) = \dots ?$ How can I can compute this last tensor product? Thanks. Also any further reading recomendation about examples of simple tensor product product calculations and not only the abstract background would be appreciated ;)