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# Linear Algebra - Span/Subspaces Watch

1. Let denote the span of a subset of a vector space V. Let S and T be subsets of V.

If I have a vector and then does this show that ?
2. Its christmas soon =/ Get pissed and stop worrying about maths
3. If then , and if, further, , then .

Think of what it means for to be a member of and what it means for it to be a member of (pardon the different notation!). Do these coincide? If so, then they're equal.
4. without writing anything what would you guess?
5. (Original post by TheEd)
If I have a vector and then does this show that ?
No, it's more like this has to be true for all , and even then, you would have just shown ; you then need to show the other direction to prove equality.
6. (Original post by nuodai)
If then , and if, further, , then .

Think of what it means for to be a member of and what it means for it to be a member of (pardon the different notation!). Do these coincide? If so, then they're equal.
So I need to show the other direction.

A different question:

If I have a vector space V and subspaces U and W of V and I want to show that V = U+W then is the following true?

If for all and then ?
7. (Original post by TheEd)

If for all and then ?
False, you are never going to have v = u + w for all possible u and w. Rather, for each v, there simply has to exist some u and w, such that v = u + w.

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Updated: December 25, 2010
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