Requirements for subspaces I am reading these two sections of text:

and

Why is there an emphasis on nonempty? Isn't that by definition always true? According to these two pieces of text, a subspace needs to be:


*

*be closed under addition

*be closed under multiplicaiton

*have a zero vector.


I can see why 6b in this following text is not a subspace... it does not contain the zero vector. But according to 4.5, it's a subspace because it satisfies the two points and is nonempty right? Something seems inconsistent to me here. According to 4.5, the set of points in 6b is a subspace right?

 A: Consider the statement of Theorem 4.5 without the assumption that $W$ be non-empty.  If $W=\varnothing$, then $W$ satisfies conditions $1.$ and $2.$ of the theorem (vacuously, of course). So if we don't specify that $W$ is nonempty, then the theorem would say that the empty set is a subspace of $V$, which is false.  Thus we need to specify that $W$ is non-empty.
The set of points in $\mathbb R^2$ satisfying $x+2y=1$ is not a subspace, for many reasons.  Pick any two points $(x_1,y_1), (x_2,y_2)$ in $\mathbb R^2$ such that $x_1+2y_1=1$ and $x_2+2y_2=1$.  Then 
$$(x_1+x_2)+2(y_1+y_2)=(x_1+2y_1)+(x_2+2y_2)=1+1\neq 1,$$
 so $(x_1+x_2,y_1+y_2)$ does not satisfy the given equation.  Hence this set is not closed under addition, and therefore it is not a subspace of $\mathbb R^2$.
A: The set of points $(x,y)\in\mathbb{R}^2$ that satisfy $x+2y=1$ is not a vector space according to Theorem $4.5$. Notice that $(1,0)$ is a solution but $2 (1,0)=(2,0)$ is not, so it is not closed under scalar multiplication. 
It follows from Theorem $4.5$ that every vector subspace must contain the zero vector because for every vector $x$ it holds that $x+(-1)x=0$.  
