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Prove that if s1 and s2 are subsets of a vectorspaceV such that

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Prove that if s1 and s2 are subsets of a vectorspaceV such that....

Homework Statement


Prove that if s1 and s2 are subsets of a vector space V such that S1 is a subset of S2,then span(S1) is a subset of span(s2). In particular, if s1 is a subset of s2 and span(s1)=V, deduce that span(s2)=V.


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The Attempt at a Solution



I came up with this, but I doubt its right. Particularly, it only applies to a finite subset. I don't know how I'd modify it to fit any subset.

Let s1,s2 be subsets of V such that s1 is a subset of s2. In cases1=s2, it is clear that span(s1)=span(s2). In case s1 does not equal s2, let x1...xn be the elements of s1. Then a1x1+a2x2+....anxn for all scalars A are in span(s1). We can write x1....xn....xk as the elements of s2. Then by definition a1x1+...anxn+....akxk are in span(s2). So span(s1) is a subset of span(s2).
 
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Answers and Replies

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CompuChip
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There is a theorem (or perhaps in your text it is even used as a definition) that the span of a set (possibly infinite) consists of finite linear combinations. So if you take any vector v from span(S1) then you can write it as a1x1+a2x2+....anxn for some scalars ai and finitely many vectors xi in S1. Now you only have to show that v is then also in span(S2). That's almost trivial though, if you note that span(S2) is a vector space by definition of span.
 
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I just checked my text, and for some reason it doesn't mention that the span is a finite set of linear combinations. But now I know.

Let s1,s2 be subsets of V such that s1 is a subset of s2. In case s1=s2, it is clear that span(s1)=span(s2). In case s1 does not equal s2, let z=a1x1+a2x2+....anxn be the set of linear combinations of the elements of s1. By definition, z is an element of span(s1). Since the span of any set is a subspace, span(s2) is closed under addition and scalar multiplication. Since x1....xn are all elements of span(s2) for some a1...an, by the definition of a subspace, x=a1x1+a2x2+....anxn is also in span(s2).

Does this work?
 
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  • #4
CompuChip
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First of all, I didn't say that the set is a finite set of linear combinations. I said it is a set of finite linear combinations, which is something completely different.

Also, z = a1x1 + ... + anxn is not a set, and it needn't contain all the elements of s1. In any case, if you want to prove that span(S1) is a subset of span(S2), you will need to make clear that you are taking any element z from span(S1). You can use that if you pick such a z, you can write it in the form you gave; rather than: if you write something in the form you gave, it is in span(S1).

Finally, a sentence like
"Since x1....xn are all elements of span(s2) for some a1...an"
is weird: why are x1 ... xn elements of span(S2) and what does a1...an have to do with them? I don't see a1...an in x1...xn?
 

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