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

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 $(T+S)(u+v)$ inst

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linear algebra - Proof of $\mathcal{L}(V,W)$ is a vector space - Mathematics Stack Exchange

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