# Writing vectors as linear combinations of bases $e_i\otimes e_j$ and $e_1\wedge e_2,e_1\wedge e_3, e_2\wedge e_3$

Write $$\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} \otimes \begin{pmatrix}2 \\ 1 \\ 1\end{pmatrix} + \begin{pmatrix}1 \\ -1 \\ 5\end{pmatrix} \otimes \begin{pmatrix}4 \\ 0 \\ 3\end{pmatrix}$$ as linear combination of the basis $e_i\otimes e_j$ of $T^2(\mathbb R^3)$,

and

write for $u:=(1,2,3),v:=(3,-2,5)\in\mathbb R^3$ the exterior product $u\wedge v\in\bigwedge^2(\mathbb R^3)$ as a linear combination of the basis vectors $e_1\wedge e_2,e_1\wedge e_3, e_2\wedge e_3$ of $\bigwedge^2(\mathbb R^3)$.

I don't know how to do calculations with the tensor product and exterior product...

## 1 Answer

Hint: Write all vectors in the following form: $\pmatrix{1\\2\\3}=e_1+2e_2+3e_3$.

For the tensor product, keep in mind that $e_1\otimes e_2$ is a linearly independent element to $e_2\otimes e_1$ (so they are as much different as they can), while for the exterior product we have $$e_1\land e_2\ +\ e_2\land e_1\ =\ 0\,.$$ That's all we have to know for this exercise.