Lines in $\mathbb{A}^3$ This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem.
Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space $\mathbb{A}^3$ (over an algebraically closed field). Show that after a linear change of coordinates in $\mathbb{A}^3$, we may assume $L_i$ does not lie in the plane $z=0$ for any $i$ and that $L_i$ is in the span of $(x_i,y_i,1)$ where the $x_i$ are pairwise distinct.
I see the second condition basically says that the lines do not lie directly over each other. It seems intuitive that we can rigidly rotate the axes so that these conditions are held, and as rotation is a linear transformation, the statement follows. I want to try to get a more solid, less handwaving proof.
 A: You need a plane $H$ through the origin containing none of the lines.  So let's first show that one exists.
Without loss of generality, we can assume that none of the lines is the $z$-axis, i.e. the line corresponding to the equations $x=y=0$.  Consider the family of hyperplanes cut out by $x + ay = 0$, where $a \in k$ .  Any two such planes intersect only at points of the form $(0,0,b)$.  Since none of the lines is the $z$-axis, we conclude that each $L_i$ lies on at most one hyperplane in this family.  Since there are infinitely many planes and finitely many $L_i$, we can find a plane containing none of the lines.  QED.
Now that you know this plane exists, by changing coordinates you can assume that it is the plane $z =0$.  Since each line goes through the origin and none are contained in $z=0$, it follows immediately that every $L_i$ must contain a point of the form $(x_i,y_i,1)$.  If some of the $x_i$ are the same, then distinctness of the lines guarantees that the corresponding $y_i$ are different.  Since we have finitely many lines, we can then choose a $C \in k$ such that the automorphism $x \rightarrow x + Cy$, $y \rightarrow y$, $z \rightarrow z$ brings our original lines to ones with the desired pairwise distinctness of the first coordinate.
