We are given two linear functions $L_1 L_2 \in L (\mathbb R^2, \mathbb R^2)$ which we define addition as $(L_1 \oplus L_2)$ to be a map $(L_1 \oplus L_2) : R^2 \to R^2$ given by the formula, $$(L_1 ⊕ L_2)(x, y) = L_1(x, y) + L_2(x, y)$$
Example, suppose $L_1$ is the liner map $L_1(x, y) = (2x + y, y)$ and $L_2$ is the linear map $L_2(x, y) = (x + y, 2x)$. Then, $(L_1 \oplus L_2) = (3x+2y, 2x+y)$.
So now going back to the title, prove that if $L_1, L_2 \in L (\mathbb R^2, \mathbb R^2)$ then $L_1 \oplus L_2) \in L (\mathbb R^2, \mathbb R^2)$ also.
Please correct me if I'm wrong, or show me how this is done.
What I am assuming we do is, say that the $\oplus$ is ordinary component addition, using the example for the $x$ (from $x$,$y$) value of the linear map $(L_1 \oplus L_2)$ we will get $3x+2y$ which is just $(2x+y + x+y)$ which I restate is ordinary addition. Because we do this for both the $x$ and $y$ value of the result, our result is therefor still in $\mathbb R^2$. or $(3x+2y, 2x+y)$ is still in $(x, y)$ format.