3
$\begingroup$

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. ...).

  1. 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?
  2. Are morphisms between projective spaces, projective linear transformation, and projective transformation (homography) different names for the same concept?

Thanks and happy holliday!

  • 0
    @Ohdur: Thanks! Are projective transformations (homographies) http://en.wikipedia.org/wiki/Homography same as projective linear transformations on a projective space, or morphisms between two projective spaces? By "morphisms between projective spaces", I meant morphisms between two projective spaces which might not be derived from the same vector spaces.2011-12-24

2 Answers 2

1

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.

0

(1) If $T$ is not injective, then it simply induces a projective map from $P(V)\setminus P(\ker T)$, which is an open set in $P(V)$, to $P(W)$. The easiest examples are provided by the very projective maps, namely, the central projections from a projective space onto a smaller subspace.

(2) The terminology is quite wild, however, the two names you mention do refer to the same thing.