$W = \begin{pmatrix} a \\ -a \\ 2a \end{pmatrix} , a \ \in \mathbb{R}$
I know I can prove this a subspace in $ \mathbb{R}^3 $easily by checking closure under addition and closure under scalar multiplication. However, can I say that $W = a\begin{pmatrix} 1\\ -1\\ 2\\ \end{pmatrix}$ and thus $W = \mathrm{span}\begin{pmatrix} 1\\ -1\\ 2\\ \end{pmatrix}$. And if W is in the span ${(v_1...v_k)}$ then it is a subspace in $\mathbb{R}^3 $ ?
You defined the set: $$W = \left\{ a \cdot \begin{pmatrix}1 \\ -1 \\ 2 \end{pmatrix} \: | \: a \in \mathbb{R} \right\}$$ Which is by definition the span of $(1,-1,2)$. Yes, it is a subspace, as you said.