Representation of GL(V) on Tensor Product of V with Dual There is a question in Jacobson that I am having some trouble with (BA II, 5.1.3). The question asks:

Let $\iota$ denote the identity map of $GL(V)$. This is a representation of $GL(V)$ acting on $V$. Consider the representation $\iota^{\ast}\otimes \iota$ acting on $V^{\ast}\bigotimes V$. Show that the set of vectors $c\in V^{\ast}\bigotimes V$ such that $(\iota^{\ast}\otimes \iota)(a)c = c$ for all $a\in GL(V)$ is a one-dimensional subspace of $V^{\ast}\bigotimes V$. Find a non-zero vector in this space.

I am having trouble with this. I don't know what $\iota^\ast$ would be (I am guessing it would map the endomorphism to that which acts in the same way on the dual basis). I can see that this is a one-dimensional subspace but I don't know how to find any nonzero vectors.
Any help is appreciated.
 A: The dual pairing $V^{\ast}\otimes V\to k$ (given by $\varphi\otimes u\mapsto \varphi(u)$) bulks up to a map
$$ V^{\ast}\otimes V\otimes V\to V: (\varphi,v,u)\mapsto \varphi(u)v, $$
and so every element of $V^{\ast}\otimes V$ may be interpreted as a linear operator on $V$. In other words, this creates a linear map $V^{\ast}\otimes V\to\mathrm{End}(V)$, in  which $\varphi\otimes v$ acts as a linear operator via the formula $(\varphi\otimes v)(u)=\varphi(u)v$. One checks this is an isomorphism with bases and dimensions.
The dual action of $\mathrm{GL}(V)$ on $V^{\ast}$ is defined so as to make $V^{\ast}\otimes V\to\mathrm{End}(V)$ a $\mathrm{GL}(V)$-equivariant map, where $\mathrm{GL}(V)$ acts on $\mathrm{End}(V)$ by conjugation. Thus, if $\varphi\in V^\ast$ and $g\in\mathrm{GL}(V)$ and $u\in V$ we have $(g\cdot \varphi)(u)=\varphi(g^{-1}u)$.
One can also call this the contragredient action. If a group $G$ acts on a set $X$, then it acts on the function space $Y^X$ via $(g\cdot f)(x)=f(g^{-1}x)$. One can interpret this as "translating the graph of $f$ along the domain by $g$," as I explain in more detail here.
This converts your problem to talking about $\mathrm{GL}(V)$-invariant elements of $\mathrm{End}(V)$, or equivalently its center, which you should know is $k\cdot I_V$. In order to write these as elements of $V^{\ast}\otimes V$ you'll have to invoke bases and dual bases (I don't know of a way around this).
