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Some technical questions about spans and bases 
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#1
Jan613, 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 


#2
Jan613, 04:37 PM

Mentor
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
Jan613, 06:16 PM

Engineering
Sci Advisor
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Thanks
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 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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