It seems like the question means to set $X=K^n$. The first condition means that $T$ is homogeneous, and the second that $T(k1+x)=T(x)$ for all $x\in X$ and $k\in K$, where $1=(1,\cdots,1)\in X=K^n$.

As rpotrie says, move to projective space $PK^{n-1}$. This is the set of lines through the origin, or the $K^n$ mod the equivalence relation that $x \sim kx$ for any $k\not=0$. Write the equivalece class of $(x_1,\cdots,x_n)$ as $[x_1,\cdots,x_n]$. As $T$ is homogeneous, it drops to a map $T:PK^{n-1}\rightarrow K^n$. The second condition is just that $T [x_1+k,\cdots,x_n+k] = T[x_1,\cdots,x_n]$ for any $k\in K$. This is equivalent to $T[0,x_2-x_1,\cdots,x_n-x_1] = T[x_1,\cdots,x_n]$.

So it seems to me that $T$ is completely determined by some map (which need satisfy no further conditions at all) $PK^{n-2}\rightarrow K^n$.