I'm very new to representation theory, so this is probably a very basic question.
Let $W_1, W_2$ be two subspaces of a $k$-vector space $V$. Let $G$ be a linear algebraic group and let $\rho: G \to \text{GL}(V)$ be a representation of $G$. Let $\rho_1: G \to \text{GL}(W_1)$, $\rho_2: G \to \text{GL}(W_2)$ be subrepresentations.
Is there a way to find a subrepresentation $$\tilde{\rho}: G \to \text{GL}(W_1+W_2)?$$
I think it would definitely be possible if $W_1+W_2$ was a direct sum, but I don't want to make this assumption.
I'm thinking that I could define it like this:
Let $w_1 + w_2 \in W_1 + W_2$ and let $g \in G$. We choose a basis $\{v_1, ..., v_n \}$ of $W_1 + W_2$ such that $\{v_1, ..., v_k \} \subset W_1$ and $\{v_{k+1}, ..., v_n \} \subset W_2$. We write
$$w_1 + w_2 = a_1 v_1+ ...+ a_n v_n, \ a_i \in k$$ and define
$$\tilde{\rho}(g)(w_1 + w_2) = \rho_1(g)(a_1 v_1)+ ... + \rho_1(g)(a_k v_k)+ \rho_2(g)(a_{k+1} v_{k+1})+ ... + \rho_2(g)(a_n v_n).$$
Could this work?