In other words, .
To show that a subset of a vector space is a subspace, you only need to show that the set is closed under addition and scalar multiplication of vectors. That is that if and are in S, so are and where is any scalar.
That can be done most conveniently, in one step, by showing that is in the same set.
Well, if is in that set then for some and some . If is in that set, similarly, .
Now, . Do you see why that is again in S?
Any vector in S is of the form . Since u is in the span of (1, 0, 2), and since v is in the span of (0, 0, 1), . That means that any vector in S can be written . You should be able to see from that what the dimension is and a basis.