# Why the proof of closure under addition in Linear Map is $(T+S)(u+v)$ instead of $(T+S)(u)$ and $(T)(u+v)$?

I am reading Linear Algebra Done Right and want to prove that $$L(V, W)$$ is a vector space. I have read the solution here:

Why the proof of closure under addition in Linear Map is $$(T+S)(u+v)$$ instead of $$(T+S)(u)$$ and $$(T)(u+v)$$?

I am very confused on when we do the sum in functions, and when we do the sum in vectors.

• You're gonna have to show us an example of what you're talking about, maybe cite the proof in question or link us – D_S Jan 13 at 3:12
• @D_S is it better now? – JOHN Jan 13 at 3:15

The answer to your questions should come from carefully using all the definitions. What does it mean for a function $$\phi: V \rightarrow W$$ to be linear? It means that
$$\phi(u+v) = \phi(u) + \phi(v)$$
$$\phi(c v) = c \phi(v)$$
for all $$u, v \in V$$ and scalars $$c$$. So in the proof, they are checking that $$\phi = T + S$$ is linear.
The definition of $$T+S$$ is it's the function which sends $$v \in V$$ to $$Tv + Sv$$ in $$W$$.