Question regarding the definition of direct sum decomposition of a representation Please bear with me. I am trying to learn representation theory of finite groups from J.P. Serre's book by myself.
Here, the author has used the word 'representation' for the homomorphism $\rho : G\rightarrow GL(V)$, as well as the vector space $V$ interchangeably. But I am a little confused about the definition of direct sum of subrepresentations.
Please correct me if I have understood it right : A $subrepresentation$ of $V$ is a subspace $W\subseteq V$ such that $\rho_g (W) \subseteq W $ $\forall g\epsilon G$,and $V$ is said to be the $direct$ $ sum$ of two subrepresentations $W,W'$ if $V=W\oplus W'$ as '$a$ $vector$ $space$', where the direct sum is $internal$.
What I am confused about is what it means for a representation to be the direct sum of two subrepresentations, where by 'representation' we mean the 'homomorphism' $\rho$ itself, i.e. the exact formulation of $\rho$ in terms of $\rho_{|W}$ and $\rho_{|W'}$. I understand that if a proper basis is chosen, then the matrix of $\rho_g$ can be written as$$[\rho_g] = \begin{pmatrix} [\rho_{g|W}] & 0\\ 0&[\rho_{g|W'}]\end{pmatrix}$$
But I want a description of $\rho_g$ in terms of $\rho_{g|W}$ and $\rho_{g|W'}$ which does not depend on the choice of a basis, e.g. if $T_1: V_1 \rightarrow W_1, T_2:V_2 \rightarrow W_2$ are two linear transformations of vector spaces, we can define the direct sum $$T_1\oplus T_2 : V_1\oplus V_2 \rightarrow W_1\oplus W_2 $$ in a canonical way, when the direct sum is understood to be $external$, without reference to a basis. I guess I am confused about the $internal$ direct sum version of it. 

If $V = W_{1} \oplus W_{2}$ then for every $v \in V$, $v = u + w$ where $u \in W_{1}$ and $w \in W_{2}$. Hence
$$ \rho_{g}(v) = \rho_{g}(u + w) = \rho_{g}(u) + \rho_{g}(w) = \rho_{g}|_{W_{1}}(u) + \rho_{g}|_{W_{2}}(w) $$
Does it clarify your doubt?
