From Wikipedia's morphisms between projective spaces:
Injective linear maps $T \in L(V,W)$ between two vector spaces $V$ and $W$ over the same field $k$ induce mappings of the corresponding projective spaces $P(V) \to P(W)$ via: $$[v] \to [T(v)],$$ where $v$ is a non-zero element of $V$ and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If $T$ is not injective, it will have a null space larger than $\{0\}$; in this case the meaning of the class of $T(v)$ is problematic if $v$ is non-zero and in the null space. ...).
- In "if $T$ is not injective, the meaning of the class of $T(v)$ is problematic if $v$ is non-zero and in the null space", I wonder what kind of problem that is?
- Are morphisms between projective spaces, projective linear transformation, and projective transformation (homography) different names for the same concept?
Thanks and happy holliday!
The null vector does not represent a valid element of a projective space. If $T$ were not injective, then there would be some $v$ which itself is non-zero but for which $T(v)$ is zero. In that case, $[v]$ is an element of $P(V)$ but $[T(v)]$ is not an element of $P(W)$, thus breaking the definition of the morphism.
Ohdur wrote most of this already in his comment, so there is nothing new in this answer, except the fact that it is technically an answer and not just a comment.