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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.

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# How to study the behavior of a particular function on a Vector Space.

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 $X \in V$.

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