I'm learning projective geometry and need help with the following exercise :
We define the lines $l_1$ and $l_2$ in $\mathbb{A}^2 = \{z = 1\} \subset \mathbb{R}^3 \setminus \{0\}$ as the intersection of $\mathbb{A}^2$ with the planes $x - y = 1$ and $x + z = 2$ respectively. Find those lines and write them as affine combinations of points.
Since I'm having difficulties for the second part of the question I'm going to share my thoughts on the first part.
When we embed $\mathbb A^2$ into $\mathbb R^3$ as the plane $z=1$, we tacitly add the latter equation to whatever planar equations we're working with. Therefore, the line given by the equation $x-y=1$ in $\mathbb A^2$ becomes the system of equations $$\begin{cases}x-y&=1\\z&=1\end{cases} \, \text{in} \, \mathbb R^3.$$
Similarly, the line given by the equation $x+z=2$ in $\mathbb A^2$ becomes the system of equations $$\begin{cases}x+z&=2\\z&=1\end{cases} \, \text{in} \, \mathbb R^3.$$
I'm not sure what the author means by "Find those lines" but this is my answer to the first part of the question. Is it correct and/or do I need to add anything else? For the second part of the question I have no idea. I know the definition of an affine combination but I don't know how to write the given lines as affine combinations of points. Any help would be much appreciated.
“Find those lines” means solve the systems of equations that you’ve set up to get a pair of points $\mathbf p$ and $\mathbf q$ in the intersection. You can then express the line as $(1-\lambda)\mathbf p+\lambda\mathbf q$, $\lambda \in \mathbb R$. In general, an affine combination is a linear combination for which the sum of the coefficients is $1$. Here, this serves to keep the resulting point on the $z=1$ plane.
On the projective plane, an affine combination isn’t enough to capture all of the points on a line. If both $\mathbf p$ and $\mathbf q$ are finite, $(1-\lambda)\mathbf p+\lambda\mathbf q$ misses the line’s point at infinity. To allow for this, you must instead use the set of linear combinations $\lambda\mathbf p+\mu\mathbf q$, with $\lambda$ and $\mu$ not both zero—the join of the two points.