Why $f$ has to be invertible?

Question

Let $V,W$ be two $(n+1)$-dim. vector spaces.

Proposition: Let $P(V), P(W)$ be $n$-dim. projective spaces. $\tau \colon P(V) \to P(W)$ is a projective transformation, if it comes from an invertible linear map $f \colon V \to W$ such that $\tau\left(\left[v\right]\right) = \left[f(v)\right]$.

Question: Why $f$ has to be invertible?

Answer 1

If $f$ is not invertible, then $\ker f$ is at least one dimensional. Choose a non-zero vector $v \in \ker f.$ Then $f(v)=0.$ In this case what is the "image" of $\tau$ at the point $[v]$?