Is there exist a notion of determinant for linear operator between space the same dimension? Let linear operator $T: E \to F$ with $\dim E = \dim F = n$. Is there exist a notion of determinant of $T$,  and/or  eigenvalue of $T$?
Note that doesn´t make sense that: $Tv=\lambda v$ because $v \notin F$. 
Thanks

First consider two vector spaces  of dimension $1$. Can we identify a unique number to an isomorphism between them? (For example consider two lines in the plane).  This requires a choice of non-zero vectors in both the spaces. 
Given a linear map $T\colon V\to W$ for vector spaces of the same dimension, one can talk of the associated map on the exterior powers $T_r\colon \wedge ^r V\to \wedge ^r W$. For $r=\dim V$ this becomes a linear map of 1-dimensional bases. Any choice of bases in $V$ and $V$ would lead to canonical choice of bases in these 1-dimensional bases and so lead to a scalar associated to the map of 1-dimensional spaces. This ois the determinant map associated to $T$.
