Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$ induced form linear transformation $f:E\rightarrow E$ with the following rule:
$(f^p)(\alpha)(u_1,u_2,\cdots,u_p)=\alpha(f(u_1),\cdots,f(u_p))\qquad ,\forall\alpha\in \Lambda^p(E) ,$
and suppose that $f^0:K\rightarrow K$ is identical map.
I want to find matrix of $f^p$. I have tried to write matrices with respect to a basis of $E$ and corresponding ones to $\Lambda^p(E)$ . I prove that matrix of $f^1=f^*$ is transpose of matrix of $f$.