I am reading through Renteln's book "Manifolds, Tensors and Forms" and in his review of Linear Algebra he defines idempotent maps as homomorphisms where say some operator $T$ has the property that $T^2=T$. Then he defines a projection as an idempotent endomorphism, where an endomorphism is a linear map from a vector space to itself.
I am ok with idempotency in the context of projection, but a bit confused about defining it in general. I presume that if $T$ is the map $T:V\rightarrow W$ then $T^2$ would be $T^2:W\rightarrow Z$. To claim that the two maps are equal requires an equivalence relationship (a particular isomorphism) between the vectors in $W$ and $Z$, no ?
But if a different isomorphism is selected between $W$ and $Z$ wouldn't that destroy the idempotency of $T$ ?
The usual definition of idempotent maps requires an endomorphism.
Maybe there is an error in the book, check out the errata http://physics.csusb.edu/~prenteln/MTF_errata.pdf, in particular for page 4, Exercise 1.9.