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Some technical questions about spans and bases

by Bipolarity
Tags: bases, spans, technical
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Bipolarity
#1
Jan6-13, 04:14 PM
P: 783
I have some technical questions about spans and bases which my textbook really does not cover very well. I would appreciate answers to them. They are not textbook problems, merely specific quesitons relating to the definitions of "span" and "basis".

1) If span(S) = V, need the elements of S be in V?
2) If S is a basis for V, need the elements of S be in V?
3) If S spans V, does S span every subspace of V?
4) If S is a basis for V, is S a basis for every subspace of V?
5) If S is linearly independent, is every subset of S also linearly independent?
6) If S is linearly dependent, is every superset of S also linearly dependent?

If I am not wrong, the answers to the questions are:
1) Yes
2) Yes
3) Yes
4) No
5) Yes
6) Yes

But I would like explicit confirmation.
All help is appreciated. Thanks.

BiP
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mfb
#2
Jan6-13, 04:37 PM
Mentor
P: 11,925
All correct. For a strict mathematical formulation, you should let everything be a part of a vector space W (can be identical to V, but then questions 1 and 2 are meaningless).

3) If S spans V, does S span every subspace of V?
Yes
The span of S contains every subspace of V as subset.
AlephZero
#3
Jan6-13, 06:16 PM
Engineering
Sci Advisor
HW Helper
Thanks
P: 7,175
I disagree about (3).

If S spans V and W is a subspace of V, then the elements of S may not all be in W, so "S spans W" might contradicts the (correct IMO) answer to (1).

Also the definition of "span" in http://en.wikipedia.org/wiki/Linear_span implies that there is only ONE space that is spanned by a given set S - namely, the intersection of all subspaces of V that contain (all the vectors in) S.


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