Is the origin itself plus a line that doesn't go through the origin considered a linear subspace?

Question

Let's focus on R2 to make it easy. A line that doesn't go through the origin isn't considered a linear subspace or a vector .
My question is, if I add artificially the origin. And construct a subgroup containing both a line and the origin (0,0). Will that BE considered a subspace together?

Answer 1

Lines that don't pass through the origin are not closed under addition. Consider the line given by $y=x+1$. It contains the points $(1,2)$ and $(2,3)$, but their sum is $(3,5)$, which is not on the given line.

There are similar problems with closure under scalar multiplication. It's not just about containing the origin; linearity means much more than that.

Answer 2

Pick any point on the line. If you scale the point by $2$, does it still belong in your set?