Tensor product/Base change/Computation/2/Exercise

We consider in ${}\mathbb {R} ^{2}$ the bases ${}{\mathfrak {v}}={\begin{pmatrix}6\\-5\end{pmatrix}},{\begin{pmatrix}-4\\7\end{pmatrix}}$ and the standard basis ${}{\mathfrak {e}}$, and we consider in ${}\mathbb {R} ^{3}$ the basis ${}{\mathfrak {x}}={\begin{pmatrix}-3\\4\\-5\end{pmatrix}},{\begin{pmatrix}2\\-1\\6\end{pmatrix}},{\begin{pmatrix}5\\2\\4\end{pmatrix}}$ and the standard basis. Determine the transformation matrix of the corresponding bases of the tensor product ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$.

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