TITLE: How to determine whether a set is a vector space or not?
QUESTION [5 upvotes]: I'm currently learning Vector Spaces and although I understand the definition of what a vector space is, I can't seem to be able to find the correct answers when doing some questions. I would even say that I'm getting some answers right by pure luck and that's defeats the purpose of mathematics. 
My problem is that I don't know the correct approach to solve these questions.
Here are a few questions which I'm trying to do but I'm not sure how to arrive to my answers:
In the questions below, I should determine whether each of the sets given is a vector space or not:

$V = \{(x,y,z)\in \mathbb{R}^3 : x-y+2z = 3\}$
$V = \{p\in P_{4}[x] : p(0) + p(1) = 0\}$
$V = \{A\in M_{3*3} : A = A^{t}\}$
$V = \{A\in M_{3*3} : AA^{t} = -I\}$

Now, all of the answers are yes except for number $4$, is a vector space.
I got $1$ wrong and $4$ wrong. I got numbers $2$ and $3$ right but I'm not sure if my way of finding the answer is right. I normally just use the definition of a Vector Space but it doesn't work all the time.
Edit: I'm not simply looking for the final answer( I already have them) but I'm more interested in understanding how to approach such questions to reach the final answer.
Edit 2:
The answers given in the memo are as follows:
1. Vector Space
2. Vector Space
3. Vector Space
4. Not a  Vector Space

Could anyone please explain how to get the answer in detail and if there is a trick to quickly find the answers?
Thanks.

REPLY [3 votes]: Note that all sets are subsets of vector spaces. Thus you have to check whether they are subspaces:

Is the zero element of the underlying vector space also in the subset?
Take two arbitrary elements from the subset. Is their sum also in the subset?
Take an element $x$ of the subset and a number $\lambda$ of the underlying field (in most cases either $\mathbb R$ or $\mathbb C$). Is $\lambda x$ also in the subset?

Only if the answer to all of the above questions is "yes" the subset is a vector space.
Example: In (1) the zero vector $(0,0,0)$ is no element of $V$ because $0-0+30\neq 0$. Thus in (1) $V$ is no vector space because the answer to the first question is "no".