Question: Let $P(U1)$ and $P(U2)$ be two non-intersecting lines in the 3-dimensional projective space $RP^3 = P(\Re^4)$. Show that $R^4$ equals the direct sum $U1 ⊕ U2$.
This is a question from my problem sheet. I am not sure I understand it quite right, though.
How do we define a line in $P(\Re^4)$? $P(\Re^4)$ should be the set of all lines through the origin, am I right? Is it the union of points $[a,b,c,d]$ which determine the lines through the origin?
Now, after that, we should define $U_1$ and $U_2$ by taking the lines through each point on the lines $P(U_1)$ and $P(U_2)$ to get $2$ planes, right?
Also, any help for the rest of the problem will be appreciated.