I am new with vector space and subspace. The question asks "is the set of all vectors (x,y) where , is in subspace

So there's 10 properties a "candidate" must satisfy to be classified as a subspace. These properties are listed alpha a-d beta e-h. I don't know if alpha and beta have some sort of special significance. In the book I'm reading there are two knew operators: vector addition and scalar multiplication. These confuse me because they appear to act in the same way as regular addition and multiplication.

Here is my attempt at the question:

this doesn't hold because (1,1)+(1,1)=(2,2) which is outside of the vector space a)

so this holds b)

this holds c)

I guess this holds, I don't know how it could not. d)

*property: for each ***u** in V, there is an element **-u** in V such that **u** (Thinking) **-u** = **0**where (Thinking) denotes vector addition

so I think this property does not hold true take for example u = 1. How do I say this in a more formal way?

If **u** is any element of V and c is any real number then c scalar multiplication with **u** is in v.

I said this does not hold if c is outside of [0,1]. Or is the idea that c has to be in [0,1] to start with?

e)

I have a simillar problem, is c>1 or c<0 then it doesn't hold.

f) g) and h) I have similar problem

So since one property doesn't hold the answer to the question is no.

I'm trying really hard here, can someone help me correct anymistakes I have?