Solving for $F(v,f)$ in Tensor $F$

[smile1]

Hello everyone

Here is the problem:

Find the value $F(v,f)$ of the tensor $F=e^1\otimes e_2 +e^2\otimes(e_1+3e_3)\in T^1_1(V)$ where $v=e_1+5e_2+4e_3, f=e^1+e^2+e^3$

Does $e^1\otimes e_2=0$ in this problem?Thanks

 

[Deveno]

I wouldn't think so, it seems to me that $e^1 \otimes e_2$ corresponds to the matrix:

$E_{12} = \begin{bmatrix}0&1&0\\0&0&0\\0&0&0 \end{bmatrix}$

which is not the 0-matrix.

That is, that:

$e^i \otimes e_j (v,u^{\ast}) = u^TE_{ij}v $, a scalar in the underlying field.

 

[smile1]

I wouldn't think so, it seems to me that $e^1 \otimes e_2$ corresponds to the matrix:

$E_{12} = \begin{bmatrix}0&1&0\\0&0&0\\0&0&0 \end{bmatrix}$

which is not the 0-matrix.

That is, that:

$e^i \otimes e_j (v,u^{\ast}) = u^TE_{ij}v $, a scalar in the underlying field.

Got it, thanks a lot:)