Definition of a Subspace. When to prove that zero vector is in the set? I was going through this PDF and was reminded of an issue I always run into as outlined below. 
A subspace for $R^n$ is any collection S of vectors in $R^n$ such that 


*

*The zero vector 0 is in S.

*If u and v are in S, then u+v is in S [closed under addition].

*If u is in S and c is scalar, then cu is in S [closed under multiplication]. 
Property 1 is only needed to ensure that S is non-empty; for non-empty S, property 1 follows from property 3, 0a = 0. 
So, when do we know we have to check if the set is nonempty? I just don't understand when we have to check for zero vector(Property 1). I just check for all three each time. 
 A: Once you have shown (2) and (3) you can either complete the proof by show that $S$ is nonempty, or by showing that $0\in S$. No matter whether you do one or the other, you then have enough to conclude that $S$ is a subspace.
There's no situation where you have to do it one way or the other. It's purely up to you, depending on what you find is easiest in the situation.

As @lulu notes in a comment, pointing out that $0\notin S$ can be a quick and easy way to show/discover that something is not a subspace.

Note that it is an extremely rare situation in mathematics to be given a particular subset of a vector space and being asked to find out whether it's a subspace or not.  Essentially the only case where this happens is in exercises in introductory linear algebra courses -- and then the point is not that you need to be particularly proficient at this rather artificial task, but simply to give you a chance to develop some intuition about the meaning of the word "subspace" (and do some work with the definition such that you'll remember it later).
A: Often noting/showing that the zero-vector in the set is the easiest way to show that the set (potential subspace) is nonempty.
A: You should verify that it is non-empty when


*

*You are feeding your proof to a computer, or a similarly literal-minded being, or

*This is your first time working with vector spaces and need to make it clear that you know and understand all the parts of the definition, or

*You've defined a fairly exotic subspace (or addition operation) and it's not that obvious which element acts as the zero vector, or that it's in your subspace.
Math is a human activity, and lots of it happens based on non-precise conventions.
