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My textbook states the following:

Let $W$ denote any subspace of $V$ that contains $S$. If $w \in \operatorname{span}(S)$, then $w$ has the form $w = c_1w_1 + c_2w_2 + \cdots + c_kw_k$ for some vectors $w_1, w_2, \ldots, w_k$ in $S$ and some scalars $c_1, c_2,\ldots, c_k$. Since $S \subseteq W$, we have $w_1, w_2, \ldots, w_k \in W$. Therefore $w = c_1w_1 + c_2w_2 + \cdots + c_kw_k$ is in $W$. Because $w$, an arbitrary vector in $\operatorname{span}(S)$, belongs to $W$, it follows that $\operatorname{span}(S) \subseteq W$.

My question specifically pertains to a single sentence:

Let $W$ denote any subspace of $V$ that contains $S$.

Why must $W$ denote a subspace here rather than a subset? It seems to me like it should be a subset since there's nothing to suggest that it's a subspace? Am I misunderstanding something?

I would greatly appreciate it if people would please take the time to clarify this.

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    A subspace is a subset with extra properties. If you just had a subset, you couldn't add/scale and be sure to remain in the set. Think of it as saying, let $W$ be a special subset. – 2017-02-14
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    @MichaelBurr thanks for the response. I think I understand now. – 2017-02-14

2 Answers 2

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The "bad thing" about $S$ is that, if $s_1, s_2\in S$, then $s_1+s_2$ doesn't have to be in $S$, and $c\cdot s_1$ doesn't have to be en $S$. Since adding vectors and multiply by scalars, you cannot actually do anything with $S$. So that's why you pick a vector subspace $W$ that contains $S$. In that case, $s_1+s_2\in W$ and $c\cdot s_1\in W$ for sure.

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If $W$ is a random subset then we have no way of knowing if $w_1+w_2$ belongs to $W$ or not. If $W$ is a subspace then we know we can find a basis $\{w_1,\ldots,w_k\}$ and write any $w \in W$ as a linear combination $c_1w_1+\cdots+c_kw_k$, where $c_i $ are scalars.