Help understanding proof for vector subspace (Hoffman and Kunze)

Question

In Hoffman and Kunze, following proof is provided for this theorem:

Theorem: A non-empty subset W of V is a subspace of V iff for each pair of vectors a,b in W and each scalar c in F the vector ca + b is again in W.

Proof. Suppose that W is a non-empty subset of V such that ca + b belongs to W for all vectors a, b in W and all scalars c in F.Since W is non-empty, there is a vector p in W, and hence (-1)p+p= 0 is in W. Then if a is any vector in W and c any scalar, the vector ca = ca + 0 is in W. In particular, (-1)a = -a is in W. Finally, if a and b are in W, then a + b = 1a + b is in W.Thus W is a subspace of V. Conversely, if W is a subspace of V, a and b are in W, and c is a scalar, certainly ca + b is in W.

What I don't get is:

1. How (-1)p+p= 0 is concluded to be in W?

2. How proving a + b = 1a + b is in W helps?

Please help in solving my above queries.

Thanks.

Answer 1

1. Take $c = -1$, $a = b = p$.
2. Recall the definition of a subspace: it's a subset containing $0$ such that, for all $a, b$ contained in it, $a + b$ is contained in it, and for all $a$ contained in it and all scalars $\lambda$, $\lambda a$ is contained in it.

Answer 2

1. In the property of the theorem, take $a=b=p$ and $c=-1$.
2. Stability for addition is one of the properties which define a subspace, along with stability for scalar multiplication.