Does a subspace containing a set of vectors also contain their span?

[Dansuer]

Homework Statement

Prove that if subspace W contain a set of vectors S, then W contain the span(S)

Homework Equations

The Attempt at a Solution

Let's take a vector $x\in span(S)$, i have to show $x\in W$ also. (*)
So since $x\in span(S)$ there are scalrs $c_1...c_n$ so that $x = c_1s_1 ...c_ns_n$ where $s_1...s_n$ are elements of S.
Let's take $s_1 = \frac{x}{c_1} - \frac{c_2}{c_1} - ...-\frac{c_n}{c_1}$ which is of course an elemtent of S.
Since $S \subseteq W$ s is an element of W also.
Since W is a vector space $c_1s_1 + c_2s_2 + ... + c_ns_n = x$ is still an element of W, so x is an element of W

I'd like a check, thanks :)

EDIT: I'm adding a part after the (*)
If x is the zero vector, then any space contains the zero vector and we are done. If x is not the zero vector then there are scalars $c_1...c_n$ where at least one is not zero, let that scalar be c_1, so that $x = c_1s_1 ...c_ns_n$ where $s_1...s_n$ . . .

 

[Some Pig]

If W contains S, since W is a subspace, any linear combinations of the vectors in S will also in W, hence W contains span(S).

 

[Dansuer]

Thank you, and your proof is even much simpler and short than mine.
But i still would like to know if mine is correct.

 

[genericusrnme]

I'd just use what Some Pig used too, if you're a subspace you contain the span of any combination of your vectors.

Regarding your proof, your last two lines are all you really need as such your proof is correct.

 

[Dansuer]

Thanks a lot

 

[genericusrnme]

No problem buddy!

 

[sunjin09]

Let's take $s_1 = \frac{x}{c_1} - \frac{c_2}{c_1} - ...-\frac{c_n}{c_1}$ which is of course an elemtent of S.
Since $S \subseteq W$ s is an element of W also.

This is wrong (and useless) since s1 is already given, not what you specify, and x is a vector and c2/c1,etc. are scalars, the subtraction is undefined

 

[Dansuer]

Yes i made a huge mistake in writing that, what i meant to say was $s_1 = \frac{x}{c_1} - \frac{c_2}{c_1}s_2 - ...-\frac{c_n}{c_1}s_n$