|Jan6-13, 04:14 PM||#1|
Some technical questions about spans and bases
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:
But I would like explicit confirmation.
All help is appreciated. Thanks.
|Jan6-13, 04:37 PM||#2|
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).
|Jan6-13, 06:16 PM||#3|
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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