[Math] How to determine whether a set is a vector space or not

linear algebravector-spaces

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:

  1. $V = \{(x,y,z)\in \mathbb{R}^3 : x-y+2z = 3\}$

  2. $V = \{p\in P_{4}[x] : p(0) + p(1) = 0\}$

  3. $V = \{A\in M_{3*3} : A = A^{t}\}$

  4. $V = \{A\in M_{3*3} : AA^{t} = -I\}$

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.

Best Answer

You want to see whether the sets are subspaces of the given vector spaces.

The first necessary condition to check is whether the zero vector belongs to the set: if not, we're done because the set is not a subspace. Note that this is not sufficient, so if the zero vector is in the set we need to do other checks.

The zero vector doesn't belong to the set in number 1, nor in the set of number 4.

For numbers 2 and 3, it's easier if you recall that the null space of a linear map is a subspace; for 2 consider $$ f\colon P_4[x]\to \mathbb{R},\qquad f(p)=p(0)+p(1) $$

For 3 consider $$ g\colon M_{3*3}\to M_{3*3},\qquad g(A)=A-A^t $$

Are these maps linear?

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