I am asking for clarifications of the basic definitions in the representation theory of a subrepresentation and a character of a subrepresentation.
Given a represenation $\rho:G \to Gl(V)$ and a subrepresentation $W \subset V$, is $\rho_V(g) = \rho_W(g)?$
I think it should be true because a subrepresentation of $(V,\rho)$ is a pair $(W, \rho)$ such that $W \subseteq V \wedge \forall g \in G, \forall x \in W, \rho(g)(x) \in W $. So $G$ is mapped to the same group of matrices. If $W$ a proper subset of $V$, then only the basis changes and $\rho_V = \rho_W$ always but $V \neq W$.
One corollary says: if $V$, $W$ are representations of $G$, then $V \cong W \iff \chi_V = \chi_W$.
By definition the character is $\chi(g) = Tr(\rho(g))$. So the subrepresentations of some representation should have the same characters. But it could be that $V \neq W$ and therefore $V \ncong W$ although $\rho_V = \rho_W$.
So where is the problem in my understanding and the definitions given?