Linear transformation proof
Well, we start with a small definition of XOR for bit strings larger than zero bits. $x$ and $y$ are bit strings with the lenght $n$ and $x_{k}$ denoting the $k^{th}$ bit (from $0$ to $lenght - 1$). $||$ represents concatenation and $\oplus$ means XOR. So we get:
  $$  x \oplus y = ((x_{0} \oplus y_{0}) || (x_{1} \oplus y_{1}) || ... || (x_{n-1} \oplus y_{n-1})) $$
Our function $T$ just returns the first 4 bits from a 6 bit string (or an arbitrarily large bit string longer or equal to 4 bits, it doesn't really matter), so we can write it as:
  $$ T(b) = (b_{0} || b_{1} || b_{2} || b_{3}) $$

Now we want to show that $T(x \oplus y) = T(x) \oplus T(y)$

How could we do that? Maybe we could insert the different definitions of our functions and see where it goes from there..?

(That's maybe not the most optimal way to answer this question, but (I think...) it works.)