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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:

enter image description 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.

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

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