Vector spaces/Finite family/Tensor product/Construction/Example

For ${}K=\mathbb {R}$ and ${}V_{1}=\mathbb {R} ^{2}$ and ${}V_{2}=\mathbb {R} ^{3}$, the elements of ${}F$ (in the sense of Definition) are linear combinations like

$4e_{({\begin{pmatrix}3\\2\end{pmatrix}},{\begin{pmatrix}6\\-1\\5\end{pmatrix}})}-5e_{({\begin{pmatrix}1\\7\end{pmatrix}},{\begin{pmatrix}3\\3\\4\end{pmatrix}})}+6e_{({\begin{pmatrix}2\\4\end{pmatrix}},{\begin{pmatrix}-4\\7\\8\end{pmatrix}})}.$

Using the standard vectors of ${}\mathbb {R} ^{2}$ and of ${}\mathbb {R} ^{3}$, this is

$4e_{(3e_{1}+2e_{2},6e_{1}-e_{2}+5e_{3})}-5e_{e_{1}+7e_{2},3e_{1}+3e_{2}+4e_{3}}+6e_{(2e_{1}+4e_{2},-4e_{1}+7e_{2}+8e_{3})}.$

Because these tuples are different, we can not simplify this expression in ${}F$. The image of this element in ${}F/U=\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$ is

$4(3e_{1}+2e_{2})\otimes (6e_{1}-e_{2}+5e_{3})-5(e_{1}+7e_{2})\otimes (3e_{1}+3e_{2}+4e_{3})+6(2e_{1}+4e_{2})\otimes (-4e_{1}+7e_{2}+8e_{3}).$

This expression can be simplified as a linear combination of the family ${}e_{i}\otimes e_{j}$, ${}i=1,2$, ${}j=1,2,3$.