Prove W is a linear subspace of

I know that W is a subspace. I just don't understand the answer my book gives.

The first step is to prove that W is not empty - For example, the zero vector is in W, so W is not empty.

The second step is to prove that for every two vectors in W, their sums are also in W. Here's where I'm a bit confused by the books answer - here's what it says -

if a =

and b =

are in W, so

.

a + b =

however,

therefore a + b

W

And they did the same type of thing for the scalar step...

Why was it enough that they only checked the first elements of the group of vectors. How do they know, for example, that

???