How is this a half plane and not a half grid?

Question

In Gilbert Strang's 4th edition, page 7, problem 1.1B we have two vectors:

$v = (1,0)$ and $w = (0,1)$.

We look for the linear combinations $cv+dw$ with restrictions.

1. $c$ is a whole number
2. $c \ge 0$

He claims that adding all vectors $cv$ and $dw$ result in a half plane. I see no way to get anything other than a half grid with vertical lines through each of the whole $x$ values. I'm afraid that I missed something about combinations.

Answer 1

You've misunderstood the solution by mixing up parts (1) and (2) of the exercises. They are separate. In the book's solution,

1. $c$ is a whole number (i.e $c \in \Bbb{N} \cup \{0\}$). By varying $c$, you get
   • $cv = (c,0)$ are equally spaced points in (1)
   • $cv + dw = (c,d)$ are parallel vertical lines
2. $c\ge0$. By varying $c$, you get
   • $cv = (c,0)$ is a half line (right half of the $x$-axis)
   • $cv + dw = (c,d)$ is a half plane (right half of the $xy$-plane)