Linear Algebra, proving subset is a subspace 
Let $W$ be a subset of vector space $V$ over $K$. $\forall \mathbf{u}, \mathbf{v} \in W,\alpha \in K, \alpha \mathbf{u} + \mathbf{v} \in W$ , show that $W$ is a subspace over $K$. Hence, show that the set of linear combinations
  $$W = \{\alpha_1 \mathbf{v}_1 + \alpha_2\mathbf{v}_2 +\ldots + \alpha_n\mathbf{v}_n, \mathbf{v}_i \in V, \alpha_i \in K,i = 1, \ldots, n\}$$
  is a subspace of $V$ over $K$.

I don't understand what $W$ contains? Is $W = \{u,v\}$? or is $W = \operatorname{Span}\{u,v\}$??
Please help, I am new to linear algebra.

Without more information about a basis for W, we can't recognise what does W contain.
But if we have more information about W and V, for example, let a basis for W to be $\{\underline{u},\underline{v}\}$ and a basis for V to be 
$\{\underline{u},\underline{v},\underline{w},\underline{x}\}$

