Define an embedding $V \subset \mathbb{P}(V \oplus \mathbb{F})$. What is the relationship between projective subspaces of $\mathbb{P}(V \oplus \mathbb{F})$ and affine subspaces of $V$?
This question is from the book "Notes on Geometry" by E.Rees.
I would like to know what $V \oplus \mathbb{F}$ means ($V$ is a vector space over $\mathbb{F}$). Can we even define this direct sum? I thought the direct sum could only be defined between vector spaces over the same field. Then is this problem just treating $\mathbb{F}$ as a vector space? For example, if $V$ is a vector space over $\mathbb{R}$ with $\dim V=n$, then $\dim (V \oplus \mathbb{R})= n+1?$
I have no idea how to approach this problem since I can't define $V \oplus \mathbb{F}$.
Can anyone give me some idea?
$\Bbb F$ is a 1D vector space over itself. Think of it as short for $\Bbb F^1$.
Recall the case of the projective line $\mathbb{FP}^1$. We identify it with $\mathbb{F}\cup\{\infty\}$ where $\infty=1/0$. This is because any 1D subspace spanned by $(x,y)$ has a unique representative $(x/y,1)$ with second coordinate $1$, except for the axis $\mathbb{F}\times\{0\}$. In other words, the two kinds of 1D subspaces are those that intersect $\mathbb{F}\times\{1\}$ in a point and the subspace $\mathbb{F}\times\{0\}$ itself.
In general, a subspace $U$ of $V\oplus\mathbb{F}$ is either a subspace of $V\times\{0\}$ or it intersects $V\times\{1\}$ in an affine subspace $W$. We can reconstruct most of $U$ by creating lines through the points $(w,1)$ for all the points $w\in W$, but this is missing any elements of $U$ which are in $V\times\{0\}$. These missing points are just set of differences $\{w_1-w_2\mid (w_1,1),(w_2,1)\in W\}$.