Let $K$ be a field, and let $V$ and $W$ be vector spaces over $K$. For a nonzero $K$-[vector space](/page/Vector%20Space) $X$, write $\mathbb{P}(X) := (X \setminus \{0\})/{\sim}$, where $x \sim y$ if and only if $y = \lambda x$ for some $\lambda \in K^\times$, and write $[x]$ for the equivalence class of $x \in X \setminus \{0\}$. Let $T: V \to W$ be an injective $K$-[linear map](/page/Linear%20Map). For all $v, v' \in V \setminus \{0\}$, if $[v] = [v']$ in $\mathbb{P}(V)$, then $T(v), T(v') \in W \setminus \{0\}$ and $[T(v)] = [T(v')]$ in $\mathbb{P}(W)$.