Tensor coordinates problem

Question

Let $B = ((1,2)^T,(1,3)^T)$ be the basis of $V=\Bbb R^2$.

Find the dual basis $B^*=(e^1,e^2)$

Find the matrix of the billinear form (tensor): $ T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$ with the respect to the canonical basis.

I suppose my tensor is given with the respect to the basis $B^*$. But how could I get the expression for $[T]_K$ (with the respect to canonical basis)?

I calculated the dual basis as $B^* = ((3,-1)^T,(-2,1)^T)$

Answer 1

The correct symbol $\otimes$ is produced with \otimes. A strategy is to first find the matrix of $T$ taken with respect to the basis $\mathcal{B}$, and then convert it to the standard basis. With respect to $\mathcal{B}$, it is clear that $$[T]_{\mathcal{B}} = \begin{pmatrix} 0 & 1 \\-1 & 2 \end{pmatrix}.$$Now, we use the tensor transformation law (that physicists love to write as that mess with indices), giving $$[T]_{{\rm std}} = A^\top [T]_{\mathcal{B}} A,$$where $A = [{\rm Id}_{\Bbb R^2}]_{{\rm std},\mathcal{B}} = [{\rm Id}_{\Bbb R^2}]_{\mathcal{B},{\rm std}}^{-1}$. Compute.

Answer 2

You probably meant $\otimes$ instead of $\oplus$. Try using the fact that the matrix elements of a tensor $M=a\otimes b$ in the canonical basis are $M_{ij}=v_i\cdot Mv_j = (a\cdot v_i)(b\cdot v_j)$ where $\cdot$ is the standard dot product and $v_k$ are the basis vectors.