I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf&ved=2ahUKEwiI3drj-NHhAhXJxIsKHQSMC9QQFjAAegQIAhAB&usg=AOvVaw0_YBJKOU-rk2J9pX3aCJyF
Fulton defines a line trough points $P=(a_1,...,a_n), Q=(b_1,...,b_n) \in \Bbb{A}^n$ to be {$(a_1+t(b_1-a_1),...,a_n+t(b_n-a_n))| t \in k$},
and a line trough points $P=[a_0:...:a_n], Q=[b_0,...,b_n] \in \Bbb{P}^n$ to be {$[\mu a_0+\lambda b_0:...:\mu a_n+\lambda b_n]| \mu, \lambda \in k$, not both zero}.
In particular I'm trying to show (in both the affine and projective case) that a line corresponds to a linear subvariety of dimension $m=1$, which Fulton defines to be a variety of the form $V=V(F_1,...,F_n)$ ($degF_i=1$) that can be mapped to $V(X_{m+1},...,X_n)$, i.e., to $V(X_2,...,X_n)$ (or ($V(X_1,...,X_n)$ in the projective case) by an affine/projective change of coordinates.
I.e. that any line is such a linear subvariety and vice versa. Note that I'm only familiar with the formalism presented in Fulton, so a more classical approach is preferred.