I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace. Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ and $ X_1 > 0 $. $S$ would not be a subspace since it violates rule 3 of the definition. A negative one times the entire matrix would produce a vector that is not in the subspace. But I would say it is quite obvious that the area where X is positive is a SUBSPACE of $\mathbb{R}^2$? It seems intuitive. Why did mathematicians choose the three rules they chose? What's the motivation?
