Single dyad, tensor

Question

$$E=\vec{e^i} \otimes \vec{e_i}$$

Multiplying scalarly by the vector $\vec{a}$ will come out: $$\vec{a}\cdot E=\vec{a}\cdot\vec{e^i} \otimes \vec{e_i} = a^i\vec{e_i}$$ how to write this schedule of vector $\vec{a}$ on the vectors of mutual base

Answer 1

If $E=E^i_je_i\otimes\varepsilon^j$ and $a=a^ke_k$ is a contravariant vector the "scalar product" $r=a\cdot E$ with respect to the metric tensor field is defined by

$$g_{ik}E^i_ja^k=E^i_ja_i=r_j$$

Where $r=r_j\varepsilon^j$ is a covariant vector and $g_{ij}$ is the metric tensor.

Now, on the other hand, if you use the the inner product for the contraction (which is not the scalar product)

$$E^i_ja^j=u^i$$

you end up with the contravariant vector $u=u^ie_i$. Notice the difference!

Edit: It just occurred to me that maybe you were wondering about the actual basis elements. So for example

$$\varepsilon^j(a^ke_k)=a^k\varepsilon^j(e_k)=a^k\delta^j_k=a^j$$