Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
 $T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
 $T(k + X) = T(X)$ for all $k \in K$ and for all $X \in V$.


If $X = (x\_1, x\_2, \ldots, x\_n)$, then $k + X = (k + x\_1, k\_ + x\_2, \ldots, k + x\_n)$.


I would like to know how to study the behavior of $T$ and its effect on the vector space $V$ be studied. 



How to study the behavior of a particular function on a Vector Space.
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) $PK^{n-2}\rightarrow K^n$.


