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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.

My questions:

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.

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A line $P(U)$ in $\mathbb RP^3$ is by definition a linear plane $U \subset \mathbb R^4$, or more precisely the projection of such a plane by the projection map $P : \mathbb R^4 \to \mathbb RP^3$. If you want, you can also use parametrization : for example a line is given by $[s:t:s+t:0]$ where $s,t \in \mathbb R$, corresponding to the plane spanned by the vectors $(1,0,1,0)$ and $(0,1,1,0)$.

Now, how can you translate on $U_1,U_2$ the hypothesis that $P(U_1) \cap P(U_2) = \emptyset$ ?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. Pertinent information should be added to the post via an edit. $\endgroup$ – user642796 May 1 '17 at 5:25

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