(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

Let's take a vector [itex]x\in span(S)[/itex], i have to show [itex]x\in W [/itex] also. (*)

So since [itex]x\in span(S)[/itex] there are scalrs [itex]c_1...c_n[/itex] so that [itex]x = c_1s_1 ...c_ns_n[/itex] where [itex]s_1...s_n[/itex] are elements of S.

Let's take [itex]s_1 = \frac{x}{c_1} - \frac{c_2}{c_1} - ...-\frac{c_n}{c_1}[/itex] which is of course an elemtent of S.

Since [itex] S \subseteq W[/itex] s is an element of W also.

Since W is a vector space [itex] c_1s_1 + c_2s_2 + ... + c_ns_n = x[/itex] 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 [itex]c_1...c_n[/itex] where at least one is not zero, let that scalar be c_1, so that [itex]x = c_1s_1 ...c_ns_n[/itex] where [itex]s_1...s_n[/itex] . . .

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# Linear algebra proof check

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