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Some technical questions about spans and bases

by Bipolarity
Tags: bases, spans, technical
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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.

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Jan6-13, 04:37 PM
P: 11,815
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?
The span of S contains every subspace of V as subset.
Jan6-13, 06:16 PM
Sci Advisor
HW Helper
P: 7,105
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 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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