I am trying to understand the following: $f: \mathbb{R}P^2 \rightarrow Mat_{\mathbb{R}}(3)$
for a map from a line $L$ in $\mathbb{R}^3$ to a matrix of orthogonal projection onto $l$.
What would such a map represent? And how to understand its image, would its form be a matrix?
We are asked to find a formula for $P = f([x])$ for any $[x] = (x^0 : x^1 : x^2)$ (I DONT want the solution) I know that [x] is the notation used for equivalence classes which are used to make the standard atlas on the projective plane:
set: $U_{i} = \{(x^0 : x^1 : x^2) | x^i \neq 0\}$
$\phi_1 : U_1 \rightarrow \mathbb{R}^3 , (x^0 : x^1 : x^2) \mapsto ({x^0 \over x^1}, {x^2\over x^1})$
However never is defined the equivalence, is there one to assume when it comes to the projective plane? ex: $x$ ~ $\lambda x$ $s.t.$ $\lambda \neq 0$
Just to confirm my understanding (hopefully) the atlas I defined above, does it mean, that for two equivalent points, they are mapped one same point?
Thank you for any explanation of what this map represents!