I've been asked to write the induced action of a matrix in $M_4(\mathbb{R})$ on $\mathbb{R}^4 \otimes \mathbb R^4$, but this terminology is unfamiliar to me. What does it mean for a matrix to induce an action on a tensor product?
Similarly, what does it mean for a matrix to induce an action on $\wedge^2 (\mathbb R^4)$?
In general, given $T:U\to U$ and $S:V\to V$, we have $T\otimes S:U\otimes V\to U\otimes V$, given by $u\otimes v\mapsto Tu\otimes Sv$.
This answers both questions, as $\wedge^2(\mathbb{R}^4)$ is a quotient of $\mathbb{R}^4\otimes \mathbb{R}^4$.