Suppose the matrix of linear map $T:E \to F$ in basis of $E$ and $F$ be like $T^{j}_{i}$; I want to compute the the tensorian elements of $T_{*}(X)$ and $T^{*}(W)$ for $X \in \otimes^{k} E$ and $W \in \otimes_{k}F$.
If $T:E \to F$ be linear map then for every : \begin{align} u_1 \otimes ... \otimes u_K \in \otimes^kE \end{align} We put :
\begin{align} T(X)= T_{*}(X)=T(u_1) \otimes ... \otimes T(u_k) \in \otimes^kF \end{align} \begin{align} T_{*}: \otimes^kE \to \otimes^kF \end{align} and $T^{t}= T^{*}:F^{*} \to E^{*}$ which $\beta \mapsto \beta \circ T$ and $T^{*} : \otimes_k F \to \otimes_k E$ which $T^{*}(\beta^1 \otimes ... \otimes \beta^k) = T^{*}(\beta^1) \otimes ... \otimes T^{*}(\beta^k)$
Could anyone hint me ?