given vectors how do you determine if it is in subspace

Question

I have no idea how to even attempt this question. I tried looking it up but I still did not understand. If you could dumb it down and explain it to me I would appreciate it a lot Thanks in advance.

Answer 1

In order for a non-empty subset of a vector space to be a subspace,you need closure under addition and scalar multiplication.

In your problem only parts $a$ and $d$ satisfy these two conditions.

Check $b$ and $c$ carefully to see why they fail to be a subspace.

Answer 2

The basic checklist to be sure that $F$ is a subspace of the $\mathbb{K}$-vector space $E$ is as follows:

• $F\subset E$
• $F$ is a vector space, that is $\forall (\lambda, \mu) \in \mathbb{K}^2, \forall (u,v)\in F^2, \lambda u + \mu v \in F$

It all narrows down to what Mohammad Riazi-Kermani said but maybe it is a bit more "dumbed down".