Proving T(U+V)=T(U)+T(V) for a polynomial transformation

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eyehategod
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i have to determine whether the function is a linear transformation. i attached a picture of the problem and of my work.

Im trying to prove T(U+V)=T(U)+T(V) where U and V are polynomials.
 

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bad choice of name. linear transformations are "god given".