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.

To check that $W$ is a vector subspace you need to check the 3 following conditions: i) $W$ is non empty (clear if $V$ is non empty), ii)if $\mathbf x \in W$ and $\mathbf y \in W$, then $\mathbf x+\mathbf y \in W$. iii)If $\alpha \in K$, and $\mathbf x \in W$, then $\alpha \mathbf x \in W$
For your second question, you need to check these three conditions again. Again (i) should hold.
For ii) and iii), observe that: for $x_i$ and $y_i$ in $K$, and $v_i$ in $V$, we have:
$(x_1 v_1+...+x_n v_n)+(y_1 v_1+...+y_n v_n)=(x_1+y_1)v_1+...+(x_n+y_n)v_n$
, which is again a linear combination of the $v_i$'s.
Now check that: for $c$ in $K$:
$c(x_1 v_1+...+x_n v_n)=cx_1 v_1+...+cx_n v_n$ is again a linear combination of the $v_i$s.
