Let $\mathrm{vect}$ be the category of vector spaces over $k$ and let $V \in \mathrm{vect}$. Can one say $\mathrm{Hom}(V, V)$ (the hom-space in the category $\mathrm{vect}$) is isomorphic to $k$?
Thanks in advance!
Let $\mathrm{vect}$ be the category of vector spaces over $k$ and let $V \in \mathrm{vect}$. Can one say $\mathrm{Hom}(V, V)$ (the hom-space in the category $\mathrm{vect}$) is isomorphic to $k$?
Thanks in advance!
Say $V$ is finite dimensional so $V\cong \oplus_{i=1}^n k_i$ where $k_i=k$. Then
$$\mathrm{Hom}(V,V)\cong \mathrm{Hom}(\oplus_{i=1}^n k_i, \oplus_{j=1}^n k_j)=\oplus_{j=1}^n \mathrm{Hom}(\oplus_{i=1}^n k_i,k_j)=$$ $$\oplus_{i=1}^n \oplus_{j=1}^n \mathrm{Hom}(k_i,k_j) \cong \oplus_{i=1}^n \oplus_{j=1}^n\mathrm{Hom}(k,k)\cong \oplus_{i=1}^n \oplus_{j=1}^n k,$$
which is not $k$ in general, but finitely many copies of it. In the infinite dimensional case this is even further from being true. Your claim is of course true when $\dim(V)=1$.