Please help me understand this:
To the best of my knowledge, the projectivization $\mathbb{P}(K^k)$ of a vector space $K^k $ is defined as the set of one-dimensional subspaces of $K^{k+1} $. OK, I say, fair enough. Then a point $p \in \mathbb{P}(K^k)$ is a one dimensional space, and if that space is spanned by $v = (v_1, v_2, ... , v_{k+1}) $, then we say the homogeneous coordinates of $p $ are $(v_1, v_2, v_3.. v_{k+1} $), and these are unique up to multiplication by scalars, so we divide by the last (or first, I've seen sometimes) coordinate.
This mental image breaks down for me when I'm given $\mathbb{P}(\bigwedge^n K^k)$, how am I to understand this space? The set of one-dimensional lines in what? $\bigwedge^n K^{k+1}$?? $\bigwedge^{n+1} K^k$?
For example, how can I write down a basis for the space I'm projectivizing? ${\bigwedge^n K^k}$ has $ {n\choose{k}} $ basis elements, so am I really think that I can just add some arbitrary vector (which doesn't have to be a wedge), and that's my space? That's doesn't seem right....
So any pointers?
Thanks
The definition of $\mathbb P(K^k)$ is the set of one-dimensional subspaces of $K^k$, not of $K^{k+1}$. The $\mathbb P$ operator takes a vector space and returns its set of one-dimensional subspaces. So $\mathbb P(\Lambda^nK^k)$ is just the one-dimensional subspaces of $\Lambda^nK^k$.
Either you've been taught the notation wrongly or you've been confused by the similar notation $\mathbb P^kK$ (the diference being that the exponent is above $\mathbb P$ not $K$). The notation $\mathbb P^kK$ denotes the projective space of dimension $k$. Since the projective space has dimension one lower than the vector space it was formed from we have that $\mathbb P^kK=\mathbb P(K^{k+1})$.